The Pilsdorff Beer Company runs a fleet of trucks along the 100 mile road from Hangtown to Dry Gulch. The trucks are old, and are apt to break down at any point along the road with equal probability. Where should the company locate a garage so as to minimize the expected distance from a typical breakdown to the garage? In other words, if [math]X[/math] is a random variable giving the location of the breakdown, measured, say, from Hangtown, and [math]b[/math] gives the location of the garage, what choice of [math]b[/math] minimizes [math]E(|X - b|)[/math]? Now suppose [math]X[/math] is not distributed uniformly over [math][0,100][/math], but instead has density function [math]f_X(x) = 2x/10,00[/math]. Then what choice of [math]b[/math] minimizes [math]E(|X - b|)[/math]?

Find [math]E(X^Y)[/math], where [math]X[/math] and [math]Y[/math] are independent random variables which are uniform on [math][0, 1][/math]. Then verify your answer by simulation.

Let [math]X[/math] be a random variable that takes on nonnegative values and has distribution function [math]F(x)[/math]. Show that

Hint: Integrate by parts. Illustrate this result by calculating [math]E(X)[/math] by this method if [math]X[/math] has an exponential distribution [math]F(x) = 1 - e^{-\lambda x}[/math] for [math]x \geq 0[/math], and [math]F(x) = 0[/math] otherwise.

Let [math]X[/math] be a continuous random variable with density function [math]f_X(x)[/math]. Show that if

then

Hint: Except on the interval [math][-1, 1][/math], the first integrand is greater than the second integrand.

Let [math]X[/math] be a random variable distributed uniformly over [math][0,20][/math]. Define a new random variable [math]Y[/math] by [math]Y = \lfloor X\rfloor[/math] (the greatest integer in [math]X[/math]). Find the expected value of [math]Y[/math]. Do the same for [math]Z = \lfloor X + .5\rfloor[/math]. Compute [math]E\bigl(|X-Y|\bigr)[/math] and [math]E\bigl(|X-Z|\bigr)[/math]. (Note that [math]Y[/math] is the value of [math]X[/math] rounded off to the nearest smallest integer, while [math]Z[/math] is the value of [math]X[/math] rounded off to the nearest integer. Which method of rounding off is better? Why?)

Assume that the lifetime of a diesel engine part is a random variable [math]X[/math] with density [math]f_X[/math]. When the part wears out, it is replaced by another with the same density. Let [math]N(t)[/math] be the number of parts that are used in time [math]t[/math]. We want to study the random variable [math]N(t)/t[/math]. Since parts are replaced on the average every [math]E(X)[/math] time units, we expect about [math]t/E(X)[/math] parts to be used in time [math]t[/math]. That is, we expect that

This result is correct but quite difficult to prove. Write a program that will allow you to specify the density [math]f_X[/math], and the time [math]t[/math], and simulate this experiment to find [math]N(t)/t[/math]. Have your program repeat the experiment 500 times and plot a bar graph for the random outcomes of [math]N(t)/t[/math]. From this data, estimate [math]E(N(t)/t)[/math] and compare this with [math]1/E(X)[/math]. In particular, do this for [math]t = 100[/math] with the following two densities:

• [math]f_X = e^{-t}[/math].
• [math]f_X = te^{-t}[/math].

Let [math]X[/math] and [math]Y[/math] be random variables. The covariance [math]\rm {Cov}(X,Y)[/math] is defined by (see Exercise)

• Show that [math]\rm {cov}(X,Y) = E(XY) - E(X)E(Y)[/math].
• Using (a), show that [math]{\rm cov}(X,Y) = 0[/math], if [math]X[/math] and [math]Y[/math] are independent. (Caution: the converse is not always true.)
• Show that [math]V(X + Y) = V(X) + V(Y) + 2{\rm cov}(X,Y)[/math].

Let [math]X[/math] and [math]Y[/math] be random variables with positive variance. The correlation of [math]X[/math] and [math]Y[/math] is defined as

• Using Exercise(c), show that
  [[math]] 0 \leq V\left( \frac X{\sigma(X)} + \frac Y{\sigma(Y)} \right) = 2(1 + \rho(X,Y))\ . [[/math]]
• Now show that
  [[math]] 0 \leq V\left( \frac X{\sigma(X)} - \frac Y{\sigma(Y)} \right) = 2(1 - \rho(X,Y))\ . [[/math]]
• Using (a) and (b), show that
  [[math]] -1 \leq \rho(X,Y) \leq 1\ . [[/math]]

Let [math]X[/math] and [math]Y[/math] be independent random variables with uniform densities in [math][0,1][/math]. Let [math]Z = X + Y[/math] and [math]W = X - Y[/math]. Find

• [math]\rho(X,Y)[/math] (see Exercise).
• [math]\rho(X,Z)[/math].
• [math]\rho(Y,W)[/math].
• [math]\rho(Z,W)[/math].

When studying certain physiological data, such as heights of fathers and sons, it is often natural to assume that these data (e.g., the heights of the fathers and the heights of the sons) are described by random variables with normal densities. These random variables, however, are not independent but rather are correlated. For example, a two-dimensional standard normal density for correlated random variables has the form

• Show that [math]X[/math] and [math]Y[/math] each have standard normal densities.
• Show that the correlation of [math]X[/math] and [math]Y[/math] (see Exercise) is [math]\rho[/math].