Intro to Stochastic Modelling Math 4320 Homework 1 Due 6th ...

Intro to Stochastic Modelling
Math 4320 Homework 1
Due 6th September
Question 1: Suppose Bill hits the target with probability 0.7, and George, independently, hits the target
with probability 0.4. Bill and George go target shooting together and shoot at the same target at the same
time.
1. Given that exactly one shot hit the target, what is the probability that is was Georges shot?
2. Given that the target is hit, what is the probability that George hit it?
Question 2: (Problem 2.2 - 3rd edition, Problem 1.2.2 - 4th edition) Let N cards carry the distinct
numbers x1, x2, .., xn. If two cards are drawn at random without replacement, show that the correlation
coefficient ρ between the numbers appearing on the two cards is −1/(N − 1).
Hint: Take X variable as first choice and Y as second choice. Clearly, Y is affected by X choice.
P r(X = xi) = 1/N, P r(Y = yi) = ∑
P r(Y = yi|X = xj)P r(X = xj)
Question 3: (Problem 2.4 - 3 rd edition, Problem 1.2.4 - 4 th edition) A fair coin is tossed until the first
time that the same side appears twice in succession. Let N be the number of tosses required.
(a) Determine the probability mass function for N.
(b) Let A be the event that N is even and B be the event that N ≤ 6. Evaluate Pr{A},Pr{B} and Pr{AB}.
Hint: Possibilities are that we end up with HH or TT. Let N=n. That means when we toss coin (n-1) times,
we get HTHTHTHTHT..... or THTHTHTHT.... Start with Pr(N=2) , Pr(N=3) and then generalize. One
Identity need to be used is :
∞∑
X k = 1
1 − X
k=0
Question 4: (Problem 2.11 - 3 rd edition, Problem 1.2.11 - 4 th edition) Random variables U and V are
independent and have the probability mass functions
j=i
pU(0) = 1
3 , pV (1) = 1
2
pU(1) = 1
3 , pV (1) = 1
2
pU(2) = 1
3
Determine the probability mass function of the sum W = U+V.
Hint: It’s a discrete case so you don’t need any formula. You just need to find possible values of W by
adding U and V and then find probability for each value of W.

Question 5: (Problem 3.4 - 3 rd edition, Problem 1.3.4 - 4 th edition) Let U be a Poisson random variable
with mean µ. Determine the expected value of the random variable V=1/(1+U).
Hint: No need to find density of V. It’s same as to find E(g(U))
Question 6: (Problem 3.7 - 3 rd edition, Problem 1.3.7 - 4 th edition) Let X and Y be independent random
variables having means µ and ν, respectively. Evaluate the convolution of their mass functions to determine
the probability distribution of their sum Z=X+Y.
Question 7: (Problem 3.12 - 3 rd edition, Problem 1.3.12 - 4 th edition) Suppose that the telephone calls
coming into a certain switchboard during a one-minute time interval follow a Poisson distribution with mean
λ = 4. If the switchboard can handle at most 6 calls per minute, what is the probability that the switchboard
will receive more calls than it can handle during a specified one-minute interval?]
Question 8: (Exercise 4.4 - 3 rd edition, Exercise 1.4.4 - 4 th edition) Twelve independent random variables,
each uniformly distributed over the interval (0,1], are added, and 6 is subtracted from the total.
Determine the mean and variance of the resulting random variable.
Question 9: (Problem 4.3 - 3 rd edition, Problem 1.4.3 - 4 th edition) Let X and Y be independent random
variables uniformly distributed over the interval [θ − 1 1
2 , θ + 2 ] for some fixed θ. Show that W = X - Y has a
distribution that is independent of θ with density function
⎧
⎨ 1 + w for − 1 ≤ w < 0
fW (w) =
⎩
1 − w
0
for 0 ≤ w ≤ 1
for |w| > 1