Suppose that the probability of suffering a side effect from a certain flu vaccine is 0.005. If 1000 persons are inoculated, find the approximate probability that
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408hw05.pdf
STAT 408 Spring 2014
Homework #5 (due Friday, February 28, by 3:00 p.m.)
No credit will be given without supporting work.
1. Suppose that the probability of suffering a side effect from a certain flu vaccine is 0.005.
If 1000 persons are inoculated, find the approximate probability that
a) At most 1 person suffers.
( use Poisson approximation )
b) 4, 5, or 6 persons suffer.
2. Suppose that the proportion of genetically modified (GMO) corn in a large shipment
is 2%. Suppose 50 kernels are randomly and independently selected for testing.
a) Find the probability that exactly 2 of these 50 kernels are GMO corn.
b) Use Poisson approximation to find the probability that exactly 2 of these 50 kernels
are GMO corn.
3 – 4. Suppose a discrete random variable X has the following probability distribution:
P ( X = 1 ) = p, P ( X = k ) = ( )
!
3ln
k
k , k = 2, 3, …
( possible values of X are positive integers: 1, 2, 3, … ).
3. a) Find the value of p that would make this a valid probability distribution.
b) Find µ X
= E ( X ) by finding the sum of the infinite series.
4. c) Find the moment-generating function of X, M X ( t ).
d) Use M X ( t ) to find µ X = E ( X ). “Hint”: The answers for (b)
and (d) should be the same.
5. Suppose a discrete random variable X has the following probability distribution:
f ( k ) = P ( X = k ) = ka , k = 2, 3, … , zero otherwise.
where a = ϕ – 1 ≈ 0.618034, where ϕ is the golden ratio.
Recall ( Homework #1 Problem 7 (a) ): this a valid probability distribution.
a) Find the moment-generating function of X, M X ( t ). For which values of t does
it exist?
b) Find E ( X ).
6 – 9. (i) Give the name of the distribution of X (if it has a name), (ii) find the values
of µ and σ 2, and (iii) calculate P ( 1 ≤ X ≤ 2 ) when the moment-generating
function of X is given by
6. a) M ( t ) = ( 0.3 + 0.7 e t ) 5. b) M ( t ) = 0.45 + 0.55 e t.
7. a) M ( t ) = t
t
e
e
7.01
3.0
−
, b) M ( t ) =
2
4.01
6.0
− t
t
e
e ,
t < – ln ( 0.7 ). t < – ln ( 0.4 ).
8. a) M ( t ) = 0.3 e t + 0.4 e
2 t + 0.2 e
3 t + 0.1 e
4 t .
b) M ( t ) = ( )∑ =
10
1
0.1
x
xt e .
9. a) M ( t ) = ( ) 13 −te
e . b) M ( t ) = e 3 t
.
10. Suppose a random variable X has the following probability density function:
f ( x ) = x
1 , 1 < x < C, zero otherwise.
a) What must the value of C be so that f ( x ) is a probability density function?
b) Find P ( X < 2 ). c) Find P ( X < 3 ). d) Find µ = E ( X ).
11. Suppose a random variable X has the following probability density function:
f ( x ) = sin x, 0 < x < 2
π , zero otherwise.
a) Find P ( X < 4
π ). b) Find µ = E ( X ).
c) Find the median of the probability distribution of X.
12. Suppose a random variable X has the following probability density function:
f ( x ) = x e x, 0 < x < 1, zero otherwise.
a) Find P ( X < 2
1 ). b) Find µ = E ( X ).
c) Find the moment-generating function of X, M X ( t ).
13 – 14. For each of the following distributions, compute P ( µ – 2 σ < X < µ + 2 σ ).
13. probability density function f ( x ) = 6 x ( 1 – x ), 0 < x < 1, zero elsewhere.
14. probability mass function f ( x ) = x
2
1 , x = 1, 2, 3, … , zero elsewhere.
