A die is rolled three times. Find the probability that the sum of the outcomes is

• greater than 9.
• an odd number.

The price of a stock on a given trading day changes according to the distribution

Find the distribution for the change in stock price after two (independent) trading days.

Let [math]X_1[/math] and [math]X_2[/math] be independent random variables with common distribution

Find the distribution of the sum [math]X_1 + X_2[/math].

In one play of a certain game you win an amount [math]X[/math] with distribution

Using the program NFoldConvolution find the distribution for your total winnings after ten (independent) plays. Plot this distribution.

Consider the following two experiments: the first has outcome [math]X[/math] taking on the values 0, 1, and 2 with equal probabilities; the second results in an (independent) outcome [math]Y[/math] taking on the value 3 with probability 1/4 and 4 with probability 3/4. Find the distribution of

• [math]Y + X[/math].
• [math]Y - X[/math].

People arrive at a queue according to the following scheme: During each minute of time either 0 or 1 person arrives. The probability that 1 person arrives is [math]p[/math] and that no person arrives is [math]q = 1 - p[/math]. Let [math]C_r[/math] be the number of customers arriving in the first [math]r[/math] minutes. Consider a Bernoulli trials process with a success if a person arrives in a unit time and failure if no person arrives in a unit time. Let [math]T_r[/math] be the number of failures before the [math]r[/math]th success.

• What is the distribution for [math]T_r[/math]?
• What is the distribution for [math]C_r[/math]?
• Find the mean and variance for the number of customers arriving in the first [math]r[/math] minutes.

• A die is rolled three times with outcomes [math]X_1[/math], [math]X_2[/math], and [math]X_3[/math]. Let [math]Y_3[/math] be the maximum of the values obtained. Show that
  [[math]] P(Y_3 \leq j) = P(X_1 \leq j)^3\ . [[/math]]
  Use this to find the distribution of [math]Y_3[/math]. Does [math]Y_3[/math] have a bell-shaped distribution?
• Now let [math]Y_n[/math] be the maximum value when [math]n[/math] dice are rolled. Find the distribution of [math]Y_n[/math]. Is this distribution bell-shaped for large values of [math]n[/math]?

A baseball player is to play in the World Series. Based upon his season play, you estimate that if he comes to bat four times in a game the number of hits he will get has a distribution

Assume that the player comes to bat four times in each game of the series.

• Let [math]X[/math] denote the number of hits that he gets in a series. Using the program NFoldConvolution, find the distribution of [math]X[/math] for each of the possible series lengths: four-game, five-game, six-game, seven-game.
• Using one of the distribution found in part (a), find the probability that his batting average exceeds .400 in a four-game series. (The batting average is the number of hits divided by the number of times at bat.)
• Given the distribution [math]p_X[/math], what is his long-term batting average?

Prove that you cannot load two dice in such a way that the probabilities for any sum from 2 to 12 are the same. (Be sure to consider the case where one or more sides turn up with probability zero.)

(Lévy^{[Notes 1]}) Assume that [math]n[/math] is an integer, not prime. Show that you can find two distributions [math]a[/math] and [math]b[/math] on the nonnegative integers such that the convolution of [math]a[/math] and [math]b[/math] is the equiprobable distribution on the set 0, 1, 2, \dots, [math]n - 1[/math]. If [math]n[/math] is prime this is not possible, but the proof is not so easy. (Assume that neither [math]a[/math] nor [math]b[/math] is concentrated at 0.)

Notes