Download Math 564 - Homework 4 1. Roll two four

Math 564 - Homework 4
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).
(b) Are X and Y independent? Are X and Z independent?
(c) Compute E(XY ) and E(XZ).
2. 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
Find the mean and variance of X. Hint: write X as the sum of two random
variables.
3. 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
E[
N
X
Xn ] = µE[N ]
n=1
Note that the number of terms in the sum is random. Hint: Condition on
the value of N .
4. Random variables X and Y take on the values 0, 1, 2, · · · n. Their joint
p.m.f. is
fX,Y (j, k) = P (X = j, Y = k) =
if j + k = n and fX,Y (j, k) = 0 if j + k 6= n.
1
n! j
p (1 − p)k
j! k!
(a) Find the marginal p.m.f.’s of X and Y .
(b) Are X and Y independent?
(c) Now suppose that their joint p.m.f. is
fX,Y (j, k) = P (X = j, Y = k) =
n!
pj q k (1 − p − q)n−j−k
j! k! (n − j − k)!
when j + k ≤ n and it is zero when j + k > n. Here p and q are positive with
p + q < 1. Now find the marginal distributions of X and Y .
5. Let X be a Poisson RV. Compute the moment generating function of X.
Recall that it is given by
MX (t) = E[etX ]
6. Suppose X is a binomial RV with n trials and probability p of success.
Recall that the moment generating function of X is
MX (t) = [et p + 1 − p]n
(a) Find E[X 3 ].
(b) Now suppose Y is another binomial random variable with m trials and
probability q of success. Assume that X and Y are independent. Let Z =
X + Y . Find the mean and variance of Z and find E[Z 3 ].
7. Let X and Y be binomial random variables with the same probability of
success, p, but different numbers of trials. Let n be the number of trials for
X and m the number for Y . Assume that X and Y are independent. Let
Z = X + Y . Find the moment generating function of Z and use it to figure
out the distribution of Z. (It should be something from our catalog.)
8. Let N be a Poisson random variable with parameter λ. We have a coin
with probability p of heads. We flip the coin N times. Let X be the number
of heads I get and Y the number of tails. (Note that N = X + Y .)
(a) Find the probability mass functions of X and Y . (They should be distributions in our catalog.)
(b) Show that X and Y are independent.
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