91Ó°ÊÓ

The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by $$ f(x)= \begin{cases}a x^{2} e^{-b x^{2}} & x \geq 0 \\ 0 & x<0\end{cases} $$ where \(b=m / 2 k T\) and \(k, T\), and \(m\) denote, respectively, Boltzmann's constant, the absolute temperature, and the mass of the molecule. Evaluate \(a\) in terms of \(b\).

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)= \begin{cases}5(1-x)^{4} & 0

A randomly chosen IQ test taker obtains a score that is approximately a normal random variable with mean 100 and standard deviation 15 . What is the probability that the test score of such a person is (a) above 125 ; (b) between 90 and \(110 ?\)

Let \(X\) be a random variable that takes on values between 0 and \(c\). That is, \(P\\{0 \leq X \leq c\\}=1\). Show that $$ \operatorname{Var}(X) \leq \frac{c^{2}}{4} $$ HINT: One approach is to first argue that $$ E\left[X^{2}\right] \leq c E[X] $$ Then use this to show that $$ \operatorname{Var}(X) \leq c^{2}[\alpha(1-\alpha)] \quad \text { where } \quad \alpha=\frac{E[X]}{c} $$

Suppose that the travel time from your home to your office is normally distributed with mean 40 minutes and standard deviation 7 minutes. If you want to be 95 percent certain that you will not be late for an office appointment at 1 P.M., what is the latest time that you should leave home?

The life of a certain type of automobile tire is normally distributed with mean 34,000 miles and standard deviation 4000 miles. (a) What is the probability that such a tire lasts over 40,000 miles? (b) What is the probability that it lasts between 30,000 and 35,000 miles? (c) Given that it has survived 30,000 miles, what is the conditional probability that it survives another 10,000 miles?

A point is chosen at random on a line segment of length \(L\). Interpret this statement and find the probability that the ratio of the shorter to the longer segment is less than \(\frac{1}{4}\).

The median of a continuous random variable having distribution function \(F\) is that value \(m\) such that \(F(m)=\frac{1}{2}\). That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of \(X\). if \(X\) is (a) uniformly distributed over \((a, b)\); (b) normal with parameters \(\mu, \sigma^{2}\); (c) exponential with rate \(\lambda\).

At a certain bank, the amount of time that a customer spends being served by a teller is an exponential random variable with mean 5 minutes. If there is a customer in service when you enter the bank, what is the probability that he or she will still be with the teller after an additional 4 minutes?

Suppose that the cumulative distribution function of the random variable \(X\) is given by $$ F(x)=1-e^{-x^{2}} \quad x>0 $$ Evaluate (a) \(P\\{X>2\\}\); (b) \(P[1