91Ó°ÊÓ

The joint probability density function of \(X\) and \(Y\) is given by $$f(x, y)=e^{-(x+y)} \quad 0 \leq x<\infty, 0 \leq y<\infty$$ Find (a) \(P\\{X< Y\\}\) and (b) \(P\\{X< a\\}\)

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 random vector \((X, Y)\) is said to be uniformly distributed over a region \(R\) in the plane if, for some constant \(c,\) its joint density is $$f(x, y)=\left\\{\begin{array}{ll}c & \text { if }(x, y) \in R \\\0 & \text { otherwise }\end{array}\right.$$ (a) Show that \(1 / c=\) area of region \(R\) Suppose that \((X, Y)\) is uniformly distributed over the square centered at (0,0) and with sides of length 2. (b) Show that \(X\) and \(Y\) are independent, with each being distributed uniformly over (-1,1). (c) What is the probability that \((X, Y)\) lies in the circle of radius 1 centered at the origin? That is, find \(P\left\\{X^{2}+Y^{2} \leq 1\right\\}.\)

Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points \(X\) and \(Y\) are independent random variables such that \(X\) is uniformly distributed over \((0, L / 2) \text { and } Y \text { is uniformly distributed over }(L / 2, L) .]\) Find the probability that the distance between the two points is greater than \(L / 3 .\)

Show that \(f(x, y)=1 / x, 0

If \(X_{1}\) and \(X_{2}\) are independent exponential random variables with respective parameters \(\lambda_{1}\) and \(\lambda_{2},\) find the distribution of \(Z=X_{1} / X_{2} .\) Also compute \(P\left\\{X_{1}

The gross weekly sales at a certain restaurant are a normal random variable with mean \(\$ 2200\) and standard deviation \(\$ 230 .\) What is the probability that (a) the total gross sales over the next 2 weeks exceeds \(\$ 5000\) (b) weekly sales exceed \(\$ 2000\) in at least 2 of the next 3 weeks? What independence assumptions have you made?

According to the U.S. National Center for Health Statistics, 25.2 percent of males and 23.6 percent of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that (a) at least 110 of these 400 people never eat breakfast; (b) the number of the women who never eat breakfast is at least as large as the number of the men who never eat breakfast.

Monthly sales are independent normal random variables with mean 100 and standard deviation 5. (a) Find the probability that exactly 3 of the next 6 months have sales greater than \(100 .\) (b) Find the probability that the total of the sales in the next 4 months is greater than \(420 .\)

The monthly worldwide average number of airplane crashes of commercial airlines is \(2.2 .\) What is the probability that there will be (a) more than 2 such accidents in the next month? (b) more than 4 such accidents in the next 2 months? (c) more than 5 such accidents in the next 3 months? Explain your reasoning!