Probability Theory Problem Set

FINAL EXAM

Show all your work. Calculators are allowed. After you finish, scan and upload your exam as a pdf file.

(1) (a) (5pt) Suppose that there are 10 red balls and 12 blue balls inside a box. Draw five balls at random without replacement. What is the probability that the second and the third balls are red, but all the other balls are blue?

(b) (5pt) Consider a factory which produces electronic components. Assume that, over a long run, 3% of the components are faulty, and that component faults are independent of one another. If this factory produces a batch of 713 components, which is the most likely number of faulty components in this batch?

(2) (a) (5pt) Let f(x) be the density function for a continuous random variable X. Suppose that

f(x) =

( c cos x if x 2 (�⇡/2, ⇡/2) 0 otherwise

for some constant c to be determined. Compute the constant c. Also, find P(�⇡/2  X  ⇡/3).

(b) (5pt) Suppose that X and Y are bivariate standard normal variables with correlation 2/3. Find an expression for P(2X +Y  5) in terms of the cumulative distribution function � of standard normal distribution.

(3) (a) (4pt) Suppose that X is a random variable with distribution

P(X = 1/4) = 2/3, P(X = 3/4) = 1/3. Suppose that, given X = x, the random variable Y is binomial(3, x)-distributed. Compute the conditional distribution of X given Y = 2.

(b) (4pt) Let X and Y have joint density

f(x, y) =

( 2x + 2y � 4xy if 0 < x < 1, 0 < y < 1 0 otherwise

.

Find fY (y|X = x) for 0 < x < 1. (4) (a) (5pt) Suppose that X and Y are independent continuous random variables, with densities fX and

fY respectively. Show that the cumulative distribution function of X + Y is given by

P(X + Y  t) = Z t

�1

Z 1

�1 fX(x)fY (y � x) dx dy =

Z t

�1

Z 1

�1 fX(x � y)fY (y) dy dx.

Use the above equation to show that the density fX+Y of X + Y is

fX+Y (t) =

Z 1

�1 fX(x)fY (t � x) dx =

Z 1

�1 fX(t � y)fY (y) dy.

(b) (4pt) Suppose that X and Y are independent uniform(0, 2) random variables. Use the result in part (a), or otherwise (such as techniques in Section 5.1), compute the density of X + Y .

(c) (2pt) Find an example of jointly continuous random variables U and V such that the marginal den- sities of U and V are both uniform(0, 1) distribution, but U + V must have a different density as in part (b).

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Problem Set 3

(5) In this problem, leave your answers in terms of the cumulative distribution function � of a standard normal random variable when needed.

Let X be a standard normal random variable. Let W be a discrete random variable independent of X. Assume that P(W = 1) = P(W = �1) = 12 . Let Y = WX. (a) (4pt) Let x 2 R. Consider the indicator Y x of the event {Y  x}. Compute the conditional

expectation E[ Y x|W ]. (b) (3pt) Use part (a) to show that Y has standard normal distribution. (c) (4pt) Compute the covariance Cov(X, Y ). Explain why X and Y do not have a bivariate normal

distribution.

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