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Math 464 - Fall 10 - Homework 5
1. Roll two four-sided dice. Let
X = number of odd dice
Y = number of even dice
Z = number of dice showing 1 or 2
So each of X, Y, Z only takes on the values 0, 1, 2.
(a) Find the joint p.m.f. of (X,Y). Find the joint p.m.f. of (X,Z). You can
give your answers in the form of 3 by 3 tables.
(b) Are X and Y independent? Are X and Z independent?
(c) Compute E(XY ) and E(XZ).
2. Let X, Y be independent random variables with
E[X] = −2,
E[X 2] = 5,
E[X 3 ] = 10,
E[X 4 ] = 50,
E[Y ] = 4,
E[Y 2 ] = 20,
E[Y 3 ] = 150,
E[Y 4 ] = 600
Let Z = X + 2Y . Find the mean and variance of Z
Let W = X − 2Y 2 . Find the mean and variance of W
3. An unfair coin has probability p of heads. I flip it until I get heads, then
I flip it some more until I get tails. Let X be the total number of flips. So
here are some possible outcomes:
HT : X = 2
THT : X = 3
HHHHT : X = 5
TTHHHHT : X = 7
(a) Find the mean and variance of X. Hint: write X as the sum of two
random variables.
(b) Now let Y be the number of heads minus the number of tails. Find the
mean and variance of Y .
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4. (Exposition) Let Xn be a sequence of discrete random variables with the
same mean µ. Let N be a random variable whose values are non-negative
integers and which is independent of each Xn . Prove that
N
X
E[
Xn ] = µE[N]
n=1
Note that the number of terms in the sum is random. Hint: Condition on
the value of N.
5. (Exposition) A coin has probability p of heads. We flip it a random
number, N, of time. N has a Poisson distribution with parameter λ and is
independent of the outcomes of the flips. Let X and Y be the number of
heads and tails respectively.
(a) Find the distributions of X and Y .
(b) Show that X and Y are independent.
6. (Exposition) Let X and Y be independent random variables. X has a
Poisson distribution with parameter λ; Y is also Poisson with parameter µ.
(a) What is the joint pmf of X, Y ?
(b) Let Z = X + Y . Show that Z has a Poisson distribution and find the
parameter. (You can do this by computing its pmf or you can use generating
functions.)
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