# Probability homework help

FINAL EXAM
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(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 densities 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) = 1
2 . Let Y = W X.
(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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