probability4

typed answers with explanation either in doc or in pdf. 

 

 strict due.  by midnight 10/01in MST

 

 for Q5, you will need to use R program http://www.r-project.org.

 I also need coding (detail procedure you operate )of it.

 

you may use this for Q5.

 # hmwk5_r.txt
# p is probability of heads
p=0.25;
for (n in 1:20) {
# seq is used to record the flips
seq=as.character();
# flip until we get H
count1<-0; got_head<-0; while (got_head==0) { count1<-count1+1; flip<-rbinom(1,1,p); if (flip==0) seq=paste(seq,"T"); if (flip==1) seq=paste(seq,"H"); if (flip==1) got_head<-1; } # now flip until we get T count2<-0; got_tail<-0; while (got_tail==0) { count2<-count2+1; flip<-rbinom(1,1,p); if (flip==0) seq=paste(seq,"T"); if (flip==1) seq=paste(seq,"H"); if (flip==0) got_tail<-1; } cat(sprintf("%s: X is %d+%d=%d \n",seq,count1,count2,count1+count2)); }  

Math 464 – Fall

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3 – 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[Y ] = −1,

E[X2] = 5, E[Y 2] = 5,

E[X3] = 10, E[Y 3] = −13,

E[X4] = 50, E[Y 4] = 73

Let Z = X + 2Y . Find the mean and variance of Z
Let W = X − 2Y 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 ] = 3.

(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.

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4. 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 .

5. Write an R program to check your answers to the previous problem when
p = 1/4. 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 flips and the
values of X for the flips. You will need to modify it to compute Y for the
flips and have it generate a much larger number of samples, say 100, 000.
One way to estimate the mean and variance of X is to use the simulation to
estimate E[X] and E[X2]. 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 its standard deviation.

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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