### Description

2. Let X and Y be bivariate normal random variables with parameters ?X =3, ?Y = 4, X = 1, Y = 2, and E[XY ] = 10.8.

(a) (5 points) Compute the correlation coefficient of X and Y .

(b) (5 points) Compute the probability P(Y < 4 | X = 5).

3. Your friendly instructor shoots free throws on a basketball court. He hits

the first and misses the second, and thereafter the probability that he hits

the next shot is equal to the proportion of shots he has hit so far.

(a) (5 points) What is the probability that he hits at least 3 of his first

5 shots?

(b) (5 points) Compute the expected number of successful shots out of

5 shots.

4. (a) (5 points) Let X be a random variable with mean 10 and variance

25. Can you conclude that the probability P(5 < X < 15) is greater

than 70 percent?

If you can, give a proof. If you cannot, give a counterexample.

(b) (5 points) Let X be a random variable with mean 50 and variance

36. Can you conclude that the probability P(38 < X < 72) is greater

than 70 percent?

If you can, give a proof. If you cannot, give a counterexample.

5. Let X be the random variable with moments given by

E[Xr] =

(

0 if r is odd;

(2021)r if r is even.

(a) (5 points) Compute the moment generating function of X.

(b) (5 points) Find the support and the probability mass function of X.

6. Let X1,X2 be bivariate random variables with support and joint pdf

SX = { (x1, x2) | 0 < x1 < x2 < 2 };

f(x1, x2) = c x21

x32

for (x1, x2) 2 S,

for some unknown c.

(a) (5 points) Compute c.

(b) (5 points) Let u1 : SX ! R and u2 : SX :! R be invertible functions

given by

u1(x1, x2) = x1

x2

; u2(x1, x2) = x1x2.

Compute the inverse function v1, v2 for u1, u2.

(c) (5 points) Let Y1 = u1(X1,X2) and Y2 = u2(X1,X2).

Compute the joint support SY . (There is no need to explain your

method.)

(d) (5 points) Compute the Jacobian associated to this transformation.

(e) (5 points) Compute the joint pdf of Y1 and Y2.

7. Let X be the exponential random variable with mean 1, and let u be the

function

u(x) = ?x2 + 10x ? 30 for all x 2 (0,1).

Let SX1 = (0, c) and SX2 = (c,1), where c is the real number such that,

the function u restricted to SX1 is invertible, and the function u restricted

to SX2 is invertible.

(a) (5 points) Find the value of c.

(b) (5 points) Let v1 be the inverse function of u restricted to SX1 , and

v2 be the inverse function of u restricted to SX2 . Compute v1 and v2,

and their domains.

(c) (5 points) Compute the support and the pdf of Y = u(X).

8. (a) (2 points) The friendly instructor went to visit the casino to play

with a biased coin, where the head shows up with probability p = 15

.

Let T be the number of games needed until the head shows up 125

times. Describe the random variable T (e.g., binomial, Poisson) with

its parameter.

(b) (3 points) Use the central limit theorem to approximate P(525 <

T 675)

(c) (2 points) Let X1, . . .X100 be independent exponential random variables

with mean = 1. Let X = X1+…+X100

100 . Describe the random

variable X with its parameter.

(d) (3 points) Use the central limit theorem to approximate P(0.985 <

X 0.99)