91Ó°ÊÓ

A system consisting of one original unit plus a spare can function for a random amount of time \(X .\) If the density of \(X\) is given (in units of months) by $$ f(x)=\left\\{\begin{array}{ll} C x e^{-x / 2} & x>0 \\ 0 & x \leq 0 \end{array}\right. $$ what is the probability that the system functions for at least 5 months?

The probability density function of \(X\), the lifetime of a certain type of electronic device (measured in hours), is given by $$ f(x)=\left\\{\begin{array}{ll} \frac{10}{x^{2}} & x>10 \\ 0 & x \leq 10 \end{array}\right. $$ (a) \(\operatorname{Find} P\\{X>20\\}\) (b) What is the cumulative distribution function of \(X ?\) (c) What is the probability that of 6 such types of devices, at least 3 will function for at least 15 hours? What assumptions are you making?

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function $$ f(x)=\left\\{\begin{array}{ll} 5(1-x)^{4} & 0

Compute \(E[X]\) if \(X\) has a density function given by (a) \(f(x)=\left\\{\begin{array}{ll}\frac{1}{4} x e^{-x / 2} & x>0 \\ 0 & \text { otherwise }\end{array}\right.\) (b) \(f(x)=\left\\{\begin{array}{ll}c\left(1-x^{2}\right) & -15 \\ 0 & x \leq 5\end{array}\right.\)

The lifetime in hours of an electronic tube is a random variable having a probability density function given by $$ f(x)=x e^{-x} \quad x \geq 0 $$ Compute the expected lifetime of such a tube.

You arrive at a bus stop at 10 A.M., knowing that the bus will arrive at some time uniformly distributed between 10 and 10: 30 (a) What is the probability that you will have to wait longer than 10 minutes? (b) If, at \(10: 15,\) the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

The salaries of physicians in a certain speciality are approximately normally distributed. If 25 percent of these physicians earn less than \(\$ 180,000\) and 25 percent earn more than \(\$ 320,000,\) approximately what fraction earn (a) less than \(\$ 200,000 ?\) (b) between \(\$ 280,000\) and \(\$ 320,000 ?\)

Let \(X\) be a normal random variable with mean 12 and variance \(4 .\) Find the value of \(c\) such that \(P\\{X>c\\}=.10\).

If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain (a) at least 50 who are in favor of the proposition; (b) between 60 and 70 inclusive who are in favor; (c) fewer than 75 in favor.

Suppose that the height, in inches, of a 25 -year-old man is a normal random variable with parameters \(\mu=71\) and \(\sigma^{2}=6.25 .\) What percentage of 25 -year-old men are taller than 6 feet, 2 inches? What percentage of men in the 6-footer club are taller than 6 feet, 5 inches?