(a) Fred visits Blotchville again. He finds that the city has installed an electronic display

at the bus stop, showing the time when the previous bus arrived. The times between

arrivals of buses are still independent Exponentials with mean 10 minutes. Fred waits

for the next bus, and then records the time between that bus and the previous bus. On

average, what length of time between buses does he see?

(b) Fred then visits Blunderville, where the times between buses are also 10 minutes on

average, and independent. Yet to his dismay, he finds that on average he has to wait more

than 1 hour for the next bus when he arrives at the bus stop! How is it possible that the

average Fred-to-bus time is greater than the average bus-to-bus time even though Fred

arrives at some time between two bus arrivals? Explain this intuitively, and construct a

specific discrete distribution for the times between buses showing that this is possible.

43. Fred and Gretchen are waiting at a bus stop in Blotchville. Two bus routes, Route 1

and Route 2, have buses that stop at this bus stop. For Route i, buses arrive according

to a Poisson process with rate

A fair die is rolled, and then a coin with probability p of Heads is flipped as many times

as the die roll says, e.g., if the result of the die roll is a 3, then the coin is flipped 3

times. Let X be the result of the die roll and Y be the number of times the coin lands

Heads.

(a) Find the joint PMF of X and Y . Are they independent?

(b) Find the marginal PMFs of X and Y .

(c) Find the conditional PMFs of Y given X = x and of Y given X = x.

6. A committee of size k is chosen from a group of n women and m men. All possible

committees of size k are equally likely. Let X and Y be the numbers of women and men

on the committee, respectively.

(a) Find the joint PMF of X and Y . Be sure to specify the support.

(b) Find the marginal PMF of X in two di?erent ways: by doing a computation using

the joint PMF, and using a story.

(c) Find the conditional PMF of Y given that X = x.

7. A stick of length L (a positive constant) is broken at a uniformly random point X.

Given that X = x, another breakpoint Y is chosen uniformly on the interval [0, x].

(a) Find the joint PDF of X and Y . Be sure to specify the support.

(b) We already know that the marginal distribution of X is Unif(0, L). Check that

marginalizing out Y from the joint PDF agrees that this is the marginal distribution of

X.

(c) We already know that the conditional distribution of Y given X = x is Unif(0, x).

Check that using the definition of conditional PDFs (in terms of joint and marginal

PDFs) agrees that this is the conditional distribution of Y given X = x.

(d) Find the marginal PDF of Y .

(e) Find the conditional PDF of X given Y = y.

8. (a) Five cards are randomly chosen from a standard deck, one at a time with replacement.

Let X, Y, Z be the numbers of chosen queens, kings, and other cards. Find the joint PMF

of X, Y, Z.

(b) Find the joint PMF of X and Y .

Hint: In summing the joint PMF of X, Y, Z over the possible values of Z, note that most

terms are 0 because of the constraint that the number of chosen cards is five.

(c) Now assume instead that the sampling is without replacement (all 5-card hands are

equally likely). Find the joint PMF of X, Y, Z.

Hint: Use the naive definition of probability.

9. Let X and Y be i.i.d. Geom(p), and N = X + Y .

(a) Find the joint PMF of X, Y, N.

(b) Find the joint PMF of X and N.

(c) Find the conditional PMF of X given N = n, and give a simple description in words

of what the result says

2. In probability and statistics, one often needs to work with random quanties that can take on any value in some range. These quantities are called continuous random variables, and are often denoted by capital letters such as X. The probability the random variable takes on a value in the range [(1, b] is denoted Pr(a g X g b) and can often be computed by integrating a special type of function called the probability density function of X. A continuous function f (:3) is the probability density function of a continuous random variable if f(.\") 2 0 for all :3, and if f; f(a:) d3: = 1. If f(:c) is the probability density function of a continuous random variable X , then Pr(aXb)=/bf(a:)d:c. (a) Verify that for every p > 0 the function 0 if 1: 12). ((1) Suppose X is a random variable with probability density function as in part (a). If Pr(10 g X g 20) = 828:1,what is the mean ofX? 6. Suppose that the probability of being tutored in Statistics is 0.55, while the probability of being tutored in Physics is 0.32, and the probability of being tutored in both Statistics and Physics is 0.09. a) What is the probability of being tutored in Statistics or Physics? b) What is the probability of not being tutored in Statistics nor Physics? c) Are the events "being tutored in Statistics" and "being tutored in Physics" mutually exclusive? Explain.1. Let X be a random variable with probability distribution f(x). What is the mean if X is discrete or continuous? 3. Let X be a random variable with probability distribution f(x). Write the expected value of the random variable g(X) if X is discrete or continuous. 5. Let X and Y be random variables with joint probability distribution f(xy). Write the mean, or expected value, of the random variable g(X,Y) if X, and Y are discrete or continuous. 6. Let X be a random variable with probability distribution f(x) and mean p (mu). Write the variance of X for when X is discrete or continuous. 9. Let X and Y be random variables with covariance oxy and standard deviations ox and gy respectively. Write the correletion coefficient of X and Y. (P.125)