Practice Exam Prob & Stats Exam 2 1

GOOD LUCK!

NAME:

PART I: Multiple Choice. Clearly circle the letter of the most correct answer to each question. Eachquestion is worth 4 points.

1. Suppose we have two random variables X and Y , and wish to have a measure of how they are relatedto each other. Which of the following is the most easily interpretable such measure?

a) Cov(X,Y )

b) E(XY )

c) ρX,Y

d) Var(XY )

2. Which of the following best describes the meaning of the word “Statistic”?

a) It is a function of random variables.

b) It is the expected value of two or more random variables.

c) It is the probability distribution of X̄.

d) It is a random sample.

3. Let X be a continuous random variable with cdf F (x). If I wanted to know the pdf of X, would Iintegrate or differentiate F (x)?

a) integrate

b) differentiate

4. Suppose X ∼ Binomial(n = 900, p = .2). Let Y = X−18012 . Then Y has approximately what distribu-

tion?

a) N(180,12)

b) N(0,1)

c) Binomial(180,.8)

d) None of the above

5. Suppose I have three discrete random variables X1, X2, and X3, each of which can have the values 0,1, 2, 3, 4, or 5 with positive probability. Which of the following is true about X̄?

a) It is a discrete random variable.

b) It is a continuous random variable.

c) It is a statistic but not a random variable.

Practice Exam Prob & Stats Exam 2 2

PART II: Problem Solving. You must show all your work to receive credit, or to receive partial creditfor incorrect answers.

1. The lifetime of a certain type of storage battery has a normal distribution with a mean of 3.0 yearsand a standard deviation of 0.5 year.

(a) (6 points) Suppose we run an experiment on just one such battery. What is the probability thatthis battery dies out in less than 2.3 years?

(b) (7 points) Suppose now we select a random sample of 3 such batteries and for each measure howlong they last. Let X1 represent the lifetime of the first battery, X2 the lifetime of the secondbattery, and X3 the lifetime of the third. Write down the joint pdf of X1, X2, and X3. Simplifyit as much as possible.

(c) (7 points) Based on the experiment in part (b), what is the probability that one of the three diesout in less than 2.3 years and the other two last over 3.7 years?

Practice Exam Prob & Stats Exam 2 3

2. Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 psi. Let X denotethe actual air pressure for the right tire and Y denote the actual air pressure for the left tire. X andY are random variables with joint pdf given by:

fX,Y (x, y) = k(x2 + y2), 30 ≤ x < 50; 30 ≤ y < 50

and equals 0 otherwise.

(a) (5 points) Find k. (NOTE: In the parts that follow, you don’t have to use any particular valuefor k if you are unsure of your answer to this part...just leave it as k.)

(b) (4 points) What is the probability that both tires are underfilled?

(c) (4 points) What is P (20 < X < 30, 40 < Y < 50)?

(d) (7 points) What is the marginal pdf of X?

Practice Exam Prob & Stats Exam 2 4

3. Our company manufactures a synthetic fabric in rolls of 10 meter lengths. For any randomly se-lected roll, the number of imperfections is a discrete random variable with the following probabilitydistribution:

x P (X = x)0 0.41 0.32 0.23 0.1

Suppose we randomly select 2 such rolls of this synthetic fabric and measure the number of imperfectionson each. Let X1 be the random variable representing the number of imperfections in the first roll, andX2 for the second roll. These two measurements are independent of each other.

(a) (5 points) What is the probability that there are no imperfections on either roll?

(b) (5 points) What is the probability that the average number of imperfections for the two rolls isgreater than or equal to 1.5?

(c) (5 points) Let Y = the maximum number of imperfections among these two rolls. Find P (Y ≥ 2).

Practice Exam Prob & Stats Exam 2 5

4. Set up the integration for each of the following problems. You do not have to solve them. Be sure toclearly show your limits and order of integration.

(a) (5 points) X ∼ Gamma(α = 3, β = 2) and Y ∼ Gamma(α = 2, β = 4). X and Y are independent.We want to calculate P (X < 2, Y > 3).

(b) (5 points) X and Y are continuous random variables with joint pdf given by fX,Y (x, y) =23 (x+

2y), 0 ≤ x ≤ 1; 0 ≤ y ≤ 1, and equals 0 otherwise. We want to calculate the probability thatthe average of X and Y is greater than 1/4.

(c) (5 points) X ∼ N(μ = 50, σ = 3) and Y ∼ Uniform(A = 4, B = 8). X and Y are independent.We want to calculate P (45 < X < 55, Y > 7).

(d) (5 points) X and Y are continuous random variables with joint pdf given by fX,Y (x, y) =6−x−y8 , 0 ≤ x ≤ 2; 2 ≤ y ≤ 4, and equals 0 otherwise. We want to calculate the proba-

bility that X is less than 1.

(e) (5 points) Using the same random variables as part (d) above, we want to calculate COV(X,Y ).

Practice Exam Prob & Stats Exam 2 6

FORMULA PAGE

e = 2.7183

If X ∼ Uniform(A,B) then f(x) = 1/(B −A), A ≤ x ≤ BE(X) = A+B

2

If X ∼ N(μ, σ) then f(x) = 1√2π

1σe−(x−μ)

2/2σ2 , −∞ < x <∞

E(X) = μ Var(X) = σ2

If X ∼ Gamma(α, β) then f(x) = 1βαΓ(α) x

α−1 e−x/β , x > 0

E(X) = αβ Var(X) = αβ2

If X ∼ Exponential(λ) then X ∼ Gamma(1, 1/λ)