91Ó°ÊÓ

Suppose that${10}^6$people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over$\left(0,{{10}^6}_{}\right)$. Let N denote the number that arrive in the first hour. Find an approximation for$P\left\{N=i\right\}$.

Each throw of an unfair die lands on each of the odd numbers $1,3,5$with probability C and on each of the even numbers with probability $2C$.

(a) Find C.

(b) Suppose that the die is tossed. Let X equal $1$if the result is an even number, and let it be $0$otherwise. Also, let Y equal $1$if the result is a number greater than three and let it be $0$otherwise. Find the joint probability mass function of X and Y. Suppose now that $12$independent tosses of the die are made.

(c) Find the probability that each of the six outcomes occurs exactly twice.

(d) Find the probability that $4$of the outcomes are either one or two, $4$are either three or four, and $4$are either five or six.

(e) Find the probability that at least $8$of the tosses land on even numbers.

Two fair dice are rolled. Find the joint probability mass function of $\mathrm{X}$and $\mathrm{Y}$when

(a) $\mathrm{X}$is the largest value obtained on any die and$\mathrm{Y}$is the sum of the values;

(b) $\mathrm{X}$is the value on the first die and $\mathrm{Y}$is the larger of the two values;

(c) $\mathrm{X}$is the smallest and $\mathrm{Y}$is the largest value obtained on the dice.

The joint probability density function of X and Y is given by f(x, y) = e^-(x+y) 0 … x < q, 0 … y < q Find

(a) P{X < Y} and

(b) P{X < a}.

Let X1, X2, X3, X4, X5 be independent continuous random variables having a common distribution function F and density function f, and set I = P{X1 < X2 < X3 < X4 < X5}

(a) Show that I does not depend on F. Hint: Write I as a five-dimensional integral and make the change of variables ui = F(xi), i = 1, ... , 5.

(b) Evaluate I.

(c) Give an intuitive explanation for your answer to (b).

Let $X1,X2,...$be a sequence of independent uniform $(0,1)$random variables. For a fixed constant c, define the random variable N by $N=min\{n:Xn>c\}$Is N independent of$X_N$? That is, does knowing the value of the first random variable that is greater than c affect the probability distribution of when this random variable occurs? Give an intuitive explanation for your answer.

A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 15 percent will purchase a plasma television set, and 40 percent will just be browsing. If 5 customers enter his store on a given day, what is the probability that he will sell exactly 2 ordinary sets and 1 plasma set on that day?

The number of people who enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?

A model proposed for NBA basketball supposes that when two teams with roughly the same record play each other, the number of points scored in a quarter by the home team minus the number scored by the visiting team is approximately a normal random variable with mean 1.5 and variance 6. In addition, the model supposes that the point differentials for the four quarters are independent. Assume that this model is correct.

(a) What is the probability that the home team wins?

(b) What is the conditional probability that the home team wins, given that it is behind by 5 points at halftime?

(c) What is the conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter?

A man and a woman agree to meet at a certain location about 12:30 p.m. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time uniformly distributed between 12:00 and 1 p.m., find the probability that the first to arrive waits no longer than 5 minutes. What is the probability that the man arrives first?