Download Math 464 - Fall 14 - Homework 5 1. Roll two six

Math 464 - Fall 14 - Homework 5
1. Roll two six-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 ] = −1,
E[Y 2 ] = 5,
E[Y 3 ] = −13,
E[Y 4 ] = 73
Let Z = 3X − Y . Find the mean and variance of Z
Let W = X 2 + Y 2 . Find the mean and variance of W
3. Let X and Y be independent random variables. Each of them has a
geometric distribution with E[X] = 2 and E[Y ] = 4.
(a) Find the joint p.m.f. of X and Y .
(b) Compute the probability that X + Y ≤ 4.
(c) Define two new random variables by W = min{X, Y } and Z = max{X, Y }.
Find the joint p.m.f. of W and Z. Hints: Consider the event {W = 3, Z = 3}.
It only occurs if {X = 3, Y = 3}. Now consider the event {W = 3, Z = 4}.
It can occurs in two ways. What are they? Finally, what can you say about
the event {W = 4, Z = 3}?
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4. I roll a six-sided die until I get a 6. Then I roll it some more until I get an
even number. Let X be the total number of rolls. So here are some possible
outcomes with the resulting value of X:
24126154: X=8
364: X=3
3463112: X=7
15466: X=5
Find the mean and variance of X. Hint: write X as the sum of two random
variables.
5. Write an R program to check your answers to the previous problem. You
should turn in your program and the output you get when you run it. To get
you started there is an R program on the web for this homework which will
generate 20 samples of the sequence of rolls and the resulting values of X.
One way to estimate the mean and variance of X is to use the simulation to
estimate E[X] and E[X 2 ]. Another way is to compute the “sample standard
deviation” of your sample of values of X. If x is a vector with the samples,
then sd(x) will compute standard deviation of x.
6. (Exposition) N is a RV with a Poisson distribution with parameter λ. A
coin has probability p of heads. We flip the coin N times. Let X and Y be
the number of heads and tails respectively. Assume that the process which
generates N is independent of the coin.
(a) Find the distributions of X and Y .
(b) Show that X and Y are independent.
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