EVALUATE THE MEAN AND VARIANCE OF
PROBABILITY DISTRIBUTION
1. Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
2. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
3. Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
4. Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
5. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
6. Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
7. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
8. Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
9. Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
10.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
11.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
12. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
13.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
14.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
15.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
16. Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
17. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
18.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
19.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
20.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
21.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
22. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
23.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
24.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
25.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
26.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
27. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
28.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
29.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
30.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
31.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
32. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
33.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
34.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
35.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
36.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
37. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
38.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
39.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
40.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
41.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
42. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
43.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
44.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
45.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
46.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
47. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
48.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
49.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
50.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
51.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
52. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
53.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
54.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
55.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
56.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
57. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
58.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
59.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
60.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
61.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
62. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
63.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
64.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
65.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
66.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
67. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
68.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
69.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
70.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
71.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
72. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
73.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
74.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
75.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
76.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
77. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
78.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
79.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
80.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
81.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
82. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
83.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
84.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
85.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
86.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
87. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
88.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
89.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
90.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
91.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
92. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
93.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
94.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
95.Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
96.Evaluate the mean and variance of the following probability distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
97. Given a continuous random variable 𝑋 with probability density function
𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
98.Evaluate the moment generating function (MGF) of a Poisson distribution
with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
99.Evaluate the 95% confidence interval for the mean of a normal distribution
with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and sample size 𝑛=
25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
100. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
101. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
102. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
103. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
104. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
105. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
106. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
107. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
108. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
109. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
110. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
111. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
112. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
113. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
114. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
115. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
116. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
117. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
118. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
119. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
120. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
121. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
122. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
123. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
124. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
125. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
126. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
127. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
128. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
129. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
130. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
131. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
132. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
133. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
134. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
135. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
136. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
137. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
138. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
139. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
140. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
141. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
142. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
143. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
144. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
145. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
146. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
147. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
148. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
149. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
150. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
151. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
152. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
153. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
154. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
155. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
156. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
157. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
158. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
159. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
160. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
161. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
162. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
163. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
164. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
165. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
166. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
167. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
168. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
169. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
170. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
171. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
172. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
173. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
174. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
175. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
176. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
177. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
178. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
179. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
180. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
181. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
182. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
183. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
184. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
185. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
186. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
187. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
188. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
189. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
190. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
191. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
192. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
193. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
194. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
195. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
196. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
197. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
198. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
199. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
200. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
201. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
202. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
203. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
204. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
205. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
206. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
207. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
208. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
209. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
210. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
211. Evaluate the mean and variance of the following probability
distribution:
𝑥
1
2
3
𝑃(𝑋=𝑥)
0.2
0.5
0.3
Solution:
Mean: 𝜇 =∑𝑥𝑖
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=1⋅0.2+2⋅0.5+3⋅0.3
=0.2+1.0+0.9
=2.1
Variance: 𝜎2=∑(𝑥𝑖−𝜇)2
3
𝑖=1 𝑃(𝑋=𝑥𝑖)
=(1−2.1)2⋅0.2+(2−2.1)2⋅0.5+(3−2.1)2⋅0.3
=1.21⋅0.2+0.01⋅0.5+0.81⋅0.3
=0.242+0.005+0.243
=0.49
212. Given a continuous random variable 𝑋 with probability density
function 𝑓(𝑥)=2𝑥 for 0≤𝑥≤1, evaluate the expected value 𝐸[𝑋].
Solution:
𝐸[𝑋]=∫ 𝑥
1
0⋅2𝑥 𝑑𝑥
=2∫ 𝑥2
1
0 𝑑𝑥
=2[𝑥3
3]0
1
=2(1
3−0)
=2
3
213. Evaluate the moment generating function (MGF) of a Poisson
distribution with parameter 𝜆.
Solution:
𝑀𝑋(𝑡)=𝐸[𝑒𝑡𝑋]
=∑𝑒𝑡𝑥
∞
𝑥=0 𝜆𝑥𝑒−𝜆
𝑥!
=𝑒−𝜆 ∑(𝜆𝑒𝑡)𝑥
𝑥!
∞
𝑥=0
=𝑒−𝜆𝑒𝜆𝑒𝑡
=𝑒𝜆(𝑒𝑡−1)
214. Evaluate the 95% confidence interval for the mean of a normal
distribution with known variance 𝜎2=4, given sample mean 𝑥‾ =10 and
sample size 𝑛=25.
Solution:
Standard Error (SE): SE =𝜎
√𝑛=2
√25=2
5=0.4
Critical value (z) for 95%CI: 𝑧 =1.96
Margin of Error (ME): ME =𝑧⋅SE =1.96⋅0.4=0.784
Confidence Interval: 𝑥‾±ME =10±0.784
=(9.216,10.784)
215. Evaluate the probability 𝑃(𝑋≤2) for a standard normal distribution.
Solution:
𝑃(𝑋≤2)=𝛷(2)
≈0.9772
