statistics exam 2pm PT TIMZONE

practice.pdf

Home >Mathematics homework help >Statistics homework help >statistics exam 2pm PT TIMZONE

PRACTICE PROBLEMS FOR THE FINAL

PSTAT 120A– Summer 2019 Professors Guo

Below is a list of practice questions for the final exam. I would suggest also going over old practice problems and your midterm to help you prepare. Any quiz, homework, or example problem has a chance of being on the exam. For more practice, I suggest you work through the review questions at the end of each chapter in Ross or another probability text.

1. Two students, A and B, are working independently on homework (not necessarily for the same class). Student A takes X = Exp(1) hours to finish his or her homework, while B takes Y = Exp(2) hours.

(a) Find the CDF of X/Y , the ratio of their problem-solving times.

(b) Find the probability that A finishes his or her homework before B does.

2. A fair coin is flipped twice. Let X be the number of heads in the two tosses, and Y be the indicator r.v. for the tosses landing the same way. That is, Y is equal to 1 if the tosses land in the same way, and 0 otherwise.

(a) Find the joint pmf of X and Y .

(b) Find the marginal pmfs of X and Y .

3. Let Y = Xβ, with X = Exp(1) and β > 0. The distribution of Y is called the Weibull distribution with parameter β. This generalizes the exponential, allowing for non-constant hazard functions. Weibull distributions are widely used in statistics, engineering, and survival analysis; there is even an 800-page book devoted to this distribution: The Weibull Distribution: A Handbook by Horst Rinne.

For this problem, let β = 3.

(a) Find P(Y > s+ t | Y > s) for s, t > 0. Does Y have the memoryless property? (b) Find the mean and variance of Y .

4. Let X and Y have joint pdf

fX,Y (x, y) =

{ c (x+ y) 0 < x < 1, 0 < y < 1

0 otherwise

(a) Find c to make this a valid joint pdf.

(b) Find the marginal pdfs of X and Y .

(d) Find the conditional pdf of Y given X = x.

(e) Find the conditional expectation E(Y | X = 14)

5. Naruto is choosing between two venues that will deliver food to his house. With probability 1/3 he will choose Ramen Mania (R), and with probability 2/3 he will choose Sushi (S). If he chooses R, 15 minutes after making the call, the remaining time it takes the food to arrive is exponentially distributed with average 10 minutes. If he orders S, 10 minutes after making the call, the time it takes the food to arrive is exponentially distributed with an average of 12 minutes. Given that he has already waited 25 minutes after calling, and the food has not arrived, what is the probability that he ordered from R?

6. A 3-sided fair die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second.

(a) Find the joint pmf of X and Y .

(b) Compute Cov(X,Y ).

7. Each throw of an unfair 6-sided die lands on each of the odd numbers 1,3,5 with probability c and on each of the even numbers with probability 2c.

(a) Find c.

(b) Suppose the die is tossed. Let X equal 1 if the result is an even number, and let it be 0 otherwise. Let Y equal 1 if the result is a number greater than 3, and let Y be 0 otherwise. Find the joint probability mass function.

(d) Are X and Y independent? Briefly justify your answer.

8. Let X and Y be i.i.d. exponential random variables with parameter λ = 1. Find the conditional cumulative distribution of X given X < Y .

Hint: Start by using conditional probability.

9. In basketball, there are four quarters in a game. Suppose that when UCSB plays another team that has roughly the same record of play, the number of points scored in a quarter by the home team minus the number scored by the visiting team is approximately a normal random variable with mean 1.5 and variance 6. In addition, suppose that the point differentials for the four quarters are independent.

Hint: Let Xi = Hi − Vi where Hi is the number of points scored by the home team in quarter i, Vi is the number of points scored by the visiting team in quarter i, and Xi is the number of points scored by the home team minus the number scored by the visiting team in quarter i.

(a) What is the probability that the home team wins?

(b) What is the conditional probability that the home teams wins, given that it is behind by 5 points at halftime?

(c) What is the conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter?

10. Let X be a continuous random variable with density function

fX(t) =

{ a t+ b −2 < t < 2 0 otherwise

(a) Find b.

(b) Find E(X). This may be in terms of a. (c) Find Var(X). This may be in terms of a.

(d) Give a brief (but reasonable) explanation why any value of a where a > b/2 would not be possible.

11. Let N be a number picked from the set {1, 2, 3, . . . , 10}, each number equally likely. Let X be 1 if N ≤ 5 and 0 otherwise. Let Y be 1 if N is even and 0 otherwise. (a) Find E[X] and E[Y ]. (b) Find Cov(X,Y ).

(d) Find E[(X + Y )2].

12. Let U1 and U2 be two independent, uniform random variables on [0, 1]. Let X = min(U1, U2), where min(u1, u2) is the smaller of two numbers u1, u2.

(a) Find the cumulative distribution function of X.

(b) Find the probability density function of X.

13. You’ve been bitten by a zombie! The cure for zombie-ism (found only recently) is administered at the Woodbury Community Clinic in Georgia. When you stagger into the clinic, you see your professor (who was also bitten by a zombie) waiting for the cure to be administered. Suppose the wait time for the cure to be administered is modeled by an exponential random variable with average wait time 20 minutes.

For the following parts, be sure to show your work (e.g. integration) and indicate where you use any rules/special properties.

(a) What is the probability that you will get the cure in the next fifteen minutes?

(b) Given that you do not know the time your professor arrived at the clinic, what is the probability that your professor will get the cure in the next seven minutes?

14. Suppose that the cumulative distribution function of the random variable X is given by:

FX(x) =

 0 x < −1 1 2 +

x 4 −1 ≤ x < 2

1 x ≥ 2

(a) Compute P (−2 < X ≤ 1). (b) Compute E[X].

15. In honor of Prince/the artist formally known as Prince, you create a playlist of 16 of his greatest hits. Let Xi be the length of the ith song (in seconds) on the playlist. Assume that each Xi is distributed by a uniform random variable between 394 and 406 seconds, independent of the length of the other songs. Use the Central Limit Theorem to approximate the probability that your playlist of 16 songs is between 6000 and 6440 seconds long.

16. The amount of time a customer spends at a certain store is modeled by an exponential random variable with mean 10 minutes. If each customer’s time at the store is independent, use the Central Limit Theorem to approximate the probability that 100 randomly selected customers spend between 950 and 1050 minutes at the store. Leave your answer in terms of the standard normal distribution Φ(x).

17. Suppose X is a continuous random variable with distribution function given by

FX(x) =

 0 x < 0

x2 0 ≤ x < 1 1 1 ≤ x

(a) Find the density fX(x) of X.

(b) Find Var(X).

18. Each year, the cryptids known only as “tubes” independently kill Americans with an average kill rate of 3 Americans per year.

(a) Using a Poisson distribution, approximate the probability that 2 or fewer Americans die next year from “tubes.”

(b) Briefly explain why a Poisson approximation here is a reasonable choice and why one might use it over the Central Limit Theorem (the normal approximation)?

19. Suppose that {Xi}∞i=1 are i.i.d. random variables with mean µ <∞ and variance σ2 <∞. Fix some ε > 0. Use the Central Limit Theorem to approximate

P ( − ε < 1

n∑ i=1

Xi − µ < ε ) .

Leave your answer in terms of the standard normal distribution Φ(a) = P (N(0, 1) ≤ a). Your answer should depend on ε, n, and σ2.

Hint: 1

n∑ i=1

Xi − µ = 1

( n∑ i=1

Xi − nµ

) .

20. Let X be a random variable with cumulative distribution function

FX(s) =

 0 s < 1 1 2 s 1 ≤ s < 2 1 2 ≤ s

(a) Let Y be the random variable defined by Y = ln(X) (here, ln is the natural log). Find the cumulative distribution function FY of Y .

(b) Find E[ln(X)].

21. Suppose that X and Y are jointly continuous random variables with joint density

fX,Y (x, y) =

{ 2e−(2x+y) 0 < x <∞, 0 < y <∞ 0 otherwise

Find Cov(X,Y ). (Make sure to justify your answer.)

22. A real number is chosen uniformly from [0, 3] and then this number is squared. Let X represent the result. (That is, if U is the number chosen from [0, 3], X = U2.)

(a) What is the cumulative distribution function of X?

(b) What is the density of X?

23. Following her true dream, your professor opens a surf shop (and school) in Puerto Rico. Sup- pose that for each i, we let Xi be the number of customers that make a purchase at your professor’s shop on day i. Assume that the collection X1, X2, X3, ... are independent and

identically distributed such that Xi d = Pois(1). Use the Central Limit Theorem (i.e., a nor-

mal approximation) to approximate the probability that at most 7 customers made a purchase

within the first 9 days. That is, approximate P ( 9∑ i=1

Xi ≤ 7 )

using the Central Limit Theorem.

You can leave your answer in terms of Φ, the CDF of a standard normal random variable.

24. Let X and Y be jointly continuous random variables with joint density

fX,Y (x, y) =

{ xe−x(y+1) x, y > 0

0 otherwise

(a) For every y > 0, find fX|Y (x | y), the conditional density of X given {Y = y}. Hint: you may use the identity

∫∞ 0 ue

−u du = 1.

(b) Find the conditional expectation E[X | Y = y]. Hint: you may use the identity ∫∞ 0 u

2e−u du = 2.

25. Let X be a geometric random variable with parameter p = 1/3 and let Y be a Poisson r.v. with parameter λ = 4. Assume X and Y are independent. A rectangle is drawn with side lengths X and Y + 1. Find the expected values of the perimeter and the area of the rectangle.

26. Suppose the random variable X has a cumulative distribution function

FX(t) =

{ t

1+t t ≥ 0 0 otherwise

(a) Find the probability density function of X.

(b) Calculate P(2 ≤ X ≤ 3). (c) Calculate E[(1 +X)2e−2X ].

27. The joint probability mass function of the random variables X and Y is given by

X Y

0 1 2

1 0 1/9 1/9

2 1/3 2/9 0

3 0 1/9 a

(a) Find a.

(b) Compute Corr(X,Y ).

(d) Find the conditional expectation E(X | Y = 1)

28. Let the random variables X, Y have joint density function

fX,Y (x, y) =

{ 3(2− x)y 0 < y < 1, y < x < 2− y 0 otherwise

(a) Find the marginal density functions fX and fY .

(b) Set up the integral to find the probability that X + Y ≤ 1. (c) Find the conditional probability density function of Y given X = x.

29. Let X be a continuous random variable with density function

fX(t) =

 a− 1 ta

t > 1

0 otherwise

Calculate the value of a such that E[X] = 2.

30. Let T be the time until a radioactive particle decays, and suppose (as if often done in physics

and chemistry) that T d = Exp(λ).

(a) The half-life of the particle is the time at which there is a 50% chance that the particle has decayed. Find the half-life of the particle.

(b) Now consider n radioactive particles with i.i.d. times until decay T1, T2, . . . Tn d = Exp(λ).

Let L be the first time at which one of the particles decays. Find the CDF of L. Hint: Note that L = min{T1, T2, · · · , Tn} and use the fact that P(L ≤ a) = 1− P(L > a).

31. Let X have density function

fX(x) =

{ 2x 0 < x < 1

0 otherwise

and let Y be uniformly distributed on the interval (0, 1). Assume X and Y are independent.

(a) Find the density function of X + Y . Hint: Start by finding the CDF of X + Y and be careful.

(b) Calculate the probability P(X + Y ≥ 3/2).

32. Let X and Y be jointly continuous random variables with joint density function

fX,Y (x, y) =

{ kx2y 0 < x < y < 1

0 otherwise

(a) Find k.

(b) Find Cov(X,Y ).

33. Suppose we roll a fair die and let X represent the number on the die.

(a) Find the moment generating function of X.

(b) Use the moment generating function to find the first two moments. That is, find E[X] and E[X2].

34. A random variable X has continuous distribution with density

fX(x) = 1

2 e−|x| for x ∈ (−∞,∞).

(a) Find the moment generating function MX(t) = E[etx]. Make sure to indicate for which t ∈ R the MGF is well-defined.

(b) Let X1, . . . , X5 be five independent and identically distributed (i.i.d.) random variables

drawn from the distribution above. If Y = 5∑ i=1

Xi, what is E[Y 2]?

Hint: Find the MGF of Y first.

35. Suppose that the moment generating function for a random variable X is given by

MX(t) = 1

(1− t)2

(a) Find a formula for the general nth moment of X. That is, in terms of n, find E[Xn]. (b) Find Var(X).

Select answers:

1. (a) 0 a < 0; a/(a+ 2) a ≥ 0 (b) 1/3

2. (a) pX,Y (1, 0) = 1/2 (b) pX(1) = 1/2, pY (1) = 1/2 (c) not independent

3. (a) Not memoryless (b) 6; 684

4. (a) 1; (b) x+1/2 ; y+1/2 (c) not independent (d) (x+y)/(x+1/2) 0 < y < 1; 0 otherwise (e) 11/18

5. 1/(1 + 2e−0.25)

6. (a) pX,Y (2,−1) = 0, pX,Y (5, 1) = 1/9 (b) 0 (c) not independent

7. (c) 6/81; (d) not independent

8. 1− e−2a a ≥ 0 ; 0 otherwise

9. (a) φ(3/ √

6) (b) 1− φ(1/ √

3) (c) φ(19/(2 √

18))

10. (a) 1/4 (b) 16a/3 (c) 4/3− (16a/3)2

11. (a) 1/2 1/2 (b) -1/20 (c) not independent (d) 14/10

12. (a) 0 a < 0; 2a− a2 0 ≤ a ≤ 1; 1 a > 1 (b) 2− 2a 0 ≤ a ≤ 1; 0 otherwise

13. (a) 1− e−0.75 (b) 1− e−0.35

14. (a) 3/4 (b) 1/8

15. φ(5/ √

3) + φ(50/ √

3)− 1

16. 2φ(1/2)− 1

17. (a) 2x 0 ≤ a ≤ 1; 0 otherwise (b) 1/18

18. (a) 17e−3/2

19. 2φ(� √ n/σ2)− 1

20. (a) 0 a < 0; 0.5ea 0 ≤ a < ln(2); 1 a ≥ ln(2) (b) ln(2)− 1/2

21. 0

22. (a) 0 a < 0; √ a/3 0 ≤ a ≤ 9; 1 a > 9 (b) 1/(6

√ a) 0 ≤ a ≤ 9; 0 otherwise

23. 1− φ(2/3)

24. (a) x(y + 1)2e−x(y+1) x > 0; 0 otherwise (b) 2/(y+1)

25. 15 16

26. (a) 1/(t+ 1)2 t > 0; 0 otherwise (b) 1/12 (c) 1/2

27. (a) 1/9 (b) 0 (c) not independent (d) 2

28. (a) fY (y) = 6y−6y2 0 < y < 1; 0 otherwise. fX(x) = 3x2(2−x)/2 0 < x < 1; 3(2−x)3/2 1 ≤ x < 2; 0 otherwise (c) 2y/x2 0 < x < 1; 2y/(2− x)2 1 ≤ x < 2; 0 otherwise

29. 3

30. (a) ln(2)/λ (b) 0 a < 0; 1− e−λnaa ≥ 0 (c) 1/(nλ)

31. (a) a2 0 < a < 1;−a2 + 2a 1 ≤ a < 2; 0 otherwise (b) 5/24

32. (a) 15 (b) 15/28 - (15/24)(15/18)

33. (a) (et + e2t + e3t + e4t + e5t + e6t)/6 (b) 21/6 91/6

34. (a) 1/(1− t2) − 1 < t < 1 (b) 10

35. (a) (n+ 1)! (b) 2