Let [math]X[/math] be a random variable with cumulative distribution function [math]F[/math] strictly increasing on the range of [math]X[/math]. Let [math]Y = F(X)[/math]. Show that [math]Y[/math] is uniformly distributed in the interval [math][0,1][/math]. (The formula [math]X = F^{-1}(Y)[/math] then tells us how to construct [math]X[/math] from a uniform random variable [math]Y[/math].)

Let [math]X[/math] be a random variable with cumulative distribution function [math]F[/math]. The median of [math]X[/math] is the value [math]m[/math] for which [math]F(m) = 1/2[/math]. Then [math]X \lt m[/math] with probability 1/2 and [math]X \gt m[/math] with probability 1/2. Find [math]m[/math] if [math]X[/math] is

• uniformly distributed over the interval [math][a,b][/math].
• normally distributed with parameters [math]\mu[/math] and [math]\sigma[/math].
• exponentially distributed with parameter [math]\lambda[/math].

Let [math]X[/math] be a random variable with density function [math]f_X[/math]. The mean of [math]X[/math] is the value [math]\mu = \int xf_x(x)\,dx[/math]. Then [math]\mu[/math] gives an average value for [math]X[/math] (see Section \ref{sec 6.3}). Find [math]\mu[/math] if [math]X[/math] is distributed uniformly, normally, or exponentially, as in Exercise.

Let [math]X[/math] be a random variable with density function [math]f_X[/math]. The mode of [math]X[/math] is the value [math]M[/math] for which [math]f(M)[/math] is maximum. Then values of [math]X[/math] near [math]M[/math] are most likely to occur. Find [math]M[/math] if [math]X[/math] is distributed normally or exponentially, as in Exercise. What happens if [math]X[/math] is distributed uniformly?

Let [math]X[/math] be a random variable normally distributed with parameters [math]\mu = 70[/math], [math]\sigma = 10[/math]. Estimate

• [math]P(X \gt 50)[/math].
• [math]P(X \lt 60)[/math].
• [math]P(X \gt 90)[/math].
• [math]P(60 \lt X \lt 80)[/math].

Bridies' Bearing Works manufactures bearing shafts whose diameters are normally distributed with parameters [math]\mu = 1[/math], [math]\sigma = .002[/math]. The buyer's specifications require these diameters to be [math]1.000 \pm .003[/math] cm. What fraction of the manufacturer's shafts are likely to be rejected? If the manufacturer improves her quality control, she can reduce the value of [math]\sigma[/math]. What value of [math]\sigma[/math] will ensure that no more than 1 percent of her shafts are likely to be rejected?

A final examination at Podunk University is constructed so that the test scores are approximately normally distributed, with parameters [math]\mu[/math] and [math]\sigma[/math]. The instructor assigns letter grades to the test scores as shown in Table (this is the process of “grading on the curve”).

What fraction of the class gets A, B, C, D, F?

(Ross^{[Notes 1]}) An expert witness in a paternity suit testifies that the length (in days) of a pregnancy, from conception to delivery, is approximately normally distributed, with parameters [math]\mu = 270[/math], [math]\sigma = 10[/math]. The defendant in the suit is able to prove that he was out of the country during the period from 290 to 240 days before the birth of the child. What is the probability that the defendant was in the country when the child was conceived?

Notes

Suppose that the time (in hours) required to repair a car is an exponentially distributed random variable with parameter [math]\lambda = 1/2[/math]. What is the probability that the repair time exceeds 4 hours? If it exceeds 4 hours what is the probability that it exceeds 8 hours?

Suppose that the number of years a car will run is exponentially distributed with parameter [math]\mu = 1/4[/math]. If Prosser buys a used car today, what is the probability that it will still run after 4 years?