Let [math]X[/math] be a random variable with [math]E(X) = \mu[/math] and [math]V(X) = \sigma^2[/math]. Show that the function [math]f(x)[/math] defined by

has its minimum value when [math]x = \mu[/math].

Let [math]X[/math] and [math]Y[/math] be two random variables defined on the finite sample space [math]\Omega[/math]. Assume that [math]X[/math], [math]Y[/math], [math]X + Y[/math], and [math]X - Y[/math] all have the same distribution. Prove that [math]P(X = Y = 0) = 1[/math].

If [math]X[/math] and [math]Y[/math] are any two random variables, then the covariance of [math]X[/math] and [math]Y[/math] is defined by [math]\rm Cov(X,Y) = E((X - E(X))(Y -E(Y)))[/math]. Note that [math]\rm Cov(X,X) = V(X)[/math]. Show that, if [math]X[/math] and [math]Y[/math] are independent, then [math]\rm Cov(X,Y) = 0[/math]; and show, by an example, that we can have [math]\rm Cov(X,Y) = 0[/math] and [math]X[/math] and [math]Y[/math] not independent.

A professor wishes to make up a true-false exam with [math]n[/math] questions. She assumes that she can design the problems in such a way that a student will answer the [math]j[/math]th problem correctly with probability [math]p_j[/math], and that the answers to the various problems may be considered independent experiments. Let [math]S_n[/math] be the number of problems that a student will get correct. The professor wishes to choose [math]p_j[/math] so that [math]E(S_n) = .7n[/math] and so that the variance of [math]S_n[/math] is as large as possible. Show that, to achieve this, she should choose [math]p_j = .7[/math] for all [math]j[/math]; that is, she should make all the problems have the same difficulty.

(Lamperti^{[Notes 1]}) An urn contains exactly 5000 balls, of which an unknown number [math]X[/math] are white and the rest red, where [math]X[/math] is a random variable with a probability distribution on the integers 0, 1, 2, ..., 5000.

• Suppose we know that [math]E(X) = \mu[/math]. Show that this is enough to allow us to calculate the probability that a ball drawn at random from the urn will be white. What is this probability?
• We draw a ball from the urn, examine its color, replace it, and then draw another. Under what conditions, if any, are the results of the two drawings independent; that is, does
  [[math]] P({{\rm white},{\rm white}}) = P({{\rm white}})^2\ ? [[/math]]
• Suppose the variance of [math]X[/math] is [math]\sigma^2[/math]. What is the probability of drawing two white balls in part (b)?

Notes

For a sequence of Bernoulli trials, let [math]X_1[/math] be the number of trials until the first success. For [math]j \geq 2[/math], let [math]X_j[/math] be the number of trials after the [math](j - 1)[/math]st success until the [math]j[/math]th success. It can be shown that [math]X_1[/math], [math]X_2[/math], ... is an independent trials process.

• What is the common distribution, expected value, and variance for [math]X_j[/math]?
• Let [math]T_n = X_1 + X_2 +\cdots+ X_n[/math]. Then [math]T_n[/math] is the time until the [math]n[/math]th success. Find [math]E(T_n)[/math] and [math]V(T_n)[/math].
• Use the results of (b) to find the expected value and variance for the number of tosses of a coin until the [math]n[/math]th occurrence of a head.

Referring to Exercise, find the variance for the number of boxes of Wheaties bought before getting half of the players' pictures and the variance for the number of additional boxes needed to get the second half of the players' pictures.

In Example, assume that the book in question has 1000 pages. Let [math]X[/math] be the number of pages with no mistakes. Show that [math]E(X) = 905[/math] and [math]V(X) = 86[/math]. Using these results, show that the probability is [math]{} \leq .05[/math] that there will be more than 924 pages without errors or fewer than 866 pages without errors.

Let [math]X[/math] be Poisson distributed with parameter [math]\lambda[/math]. Show that [math]V(X) = \lambda[/math].