The Pilsdorff beer company runs a fleet of trucks along the 100 mile road from Hangtown to Dry Gulch, and maintains a garage halfway in between. Each of the trucks is apt to break down at a point [math]X[/math] miles from Hangtown, where [math]X[/math] is a random variable uniformly distributed over [math][0,100][/math].

• Find a lower bound for the probability [math]P(|X - 50| \leq 10)[/math].
• Suppose that in one bad week, 20 trucks break down. Find a lower bound for the probability [math]P(|A_{20} - 50| \leq 10)[/math], where [math]A_{20}[/math] is the average of the distances from Hangtown at the time of breakdown.

A share of common stock in the Pilsdorff beer company has a price [math]Y_n[/math] on the [math]n[/math]th business day of the year. Finn observes that the price change [math]X_n = Y_{n + 1} - Y_n[/math] appears to be a random variable with mean [math]\mu = 0[/math] and variance [math]\sigma^2 =1/4[/math]. If [math]Y_1 = 30[/math], find a lower bound for the following probabilities, under the assumption that the [math]X_n[/math]'s are mutually independent.

• [math]P(25 \leq Y_2 \leq 35)[/math].
• [math]P(25 \leq Y_{11} \leq 35)[/math].
• [math]P(25 \leq Y_{101} \leq 35)[/math].

Suppose one hundred numbers [math]X_1[/math], [math]X_2[/math], \dots, [math]X_{100}[/math] are chosen independently at random from [math][0,20][/math]. Let [math]S = X_1 + X_2 +\cdots+ X_{100}[/math] be the sum, [math]A = S/100[/math] the average, and [math]S^* = (S - 1000)/(10/\sqrt3)[/math] the standardized sum. Find lower bounds for the probabilities

• [math]P(|S - 1000| \leq 100)[/math].
• [math]P(|A - 10| \leq 1)[/math].
• [math]P(|S^*| \leq \sqrt3)[/math].

Let [math]X[/math] be a continuous random variable normally distributed on [math](-\infty,+\infty)[/math] with mean 0 and variance 1. Using the normal table provided in Appendix A, or the program NormalArea, find values for the function [math]f(x) = P(|X| \geq x)[/math] as [math]x[/math] increases from 0 to 4.0 in steps of .25. Note that for [math]x \geq 0[/math] the table gives [math] NA(0,x) = P(0 \leq X \leq x)[/math] and thus [math]P(|X| \geq x) = 2(.5 - NA(0,x)[/math]. Plot by hand the graph of [math]f(x)[/math] using these values, and the graph of the Chebyshev function [math]g(x) = 1/x^2[/math], and compare (see Exercise Exercise).

Repeat Exercise, but this time with mean 10 and variance 3. Note that the table in Appendix A presents values for a standard normal variable. Find the standardized version [math]X^*[/math] for [math]X[/math], find values for [math]f^*(x) = P(|X^*| \geq x)[/math] as in Exercise, and then rescale these values for [math]f(x) = P(|X -10| \geq x)[/math]. Graph and compare this function with the Chebyshev function [math]g(x) = 3/x^2[/math].

Let [math]Z = X/Y[/math] where [math]X[/math] and [math]Y[/math] have normal densities with mean 0 and standard deviation 1. Then it can be shown that [math]Z[/math] has a Cauchy density.

• Write a program to illustrate this result by plotting a bar graph of 1000 samples obtained by forming the ratio of two standard normal outcomes. Compare your bar graph with the graph of the Cauchy density. Depending upon which computer language you use, you may or may not need to tell the computer how to simulate a normal random variable. A method for doing this was described in Important Densities.
• We have seen that the Law of Large Numbers does not apply to the Cauchy density (see Example). Simulate a large number of experiments with Cauchy density and compute the average of your results. Do these averages seem to be approaching a limit? If so can you explain why this might be?

Show that, if [math]X \geq 0[/math], then [math]P(X \geq a) \leq E(X)/a[/math].

(Lamperti^{[Notes 1]}) Let [math]X[/math] be a non-negative random variable. What is the best upper bound you can give for [math]P(X \geq a)[/math] if you know

• [math]E(X) = 20[/math].
• [math]E(X) = 20[/math] and [math]V(X) = 25[/math].
• [math]E(X) = 20[/math], [math]V(X) = 25[/math], and [math]X[/math] is symmetric about its mean.

Notes