For which of the following random variables would it be appropriate to assign a uniform distribution?

• Let [math]X[/math] represent the roll of one die.
• Let [math]X[/math] represent the number of heads obtained in three tosses of a coin.
• A roulette wheel has 38 possible outcomes: 0, 00, and 1 through 36. Let [math]X[/math] represent the outcome when a roulette wheel is spun.
• Let [math]X[/math] represent the birthday of a randomly chosen person.
• Let [math]X[/math] represent the number of tosses of a coin necessary to achieve a head for the first time.

Let [math]n[/math] be a positive integer. Let [math]S[/math] be the set of integers between 1 and [math]n[/math]. Consider the following process: We remove a number from [math]S[/math] at random and write it down. We repeat this until [math]S[/math] is empty. The result is a permutation of the integers from 1 to [math]n[/math]. Let [math]X[/math] denote this permutation. Is [math]X[/math] uniformly distributed?

Let [math]X[/math] be a random variable which can take on countably many values. Show that [math]X[/math] cannot be uniformly distributed.

Suppose we are attending a college which has 3000 students. We wish to choose a subset of size 100 from the student body. Let [math]X[/math] represent the subset, chosen using the following possible strategies. For which strategies would it be appropriate to assign the uniform distribution to [math]X[/math]? If it is appropriate, what probability should we assign to each outcome?

• Take the first 100 students who enter the cafeteria to eat lunch.
• Ask the Registrar to sort the students by their Social Security number, and then take the first 100 in the resulting list.
• Ask the Registrar for a set of cards, with each card containing the name of exactly one student, and with each student appearing on exactly one card. Throw the cards out of a third-story window, then walk outside and pick up the first 100 cards that you find.

Under the same conditions as in the preceding exercise, can you describe a procedure which, if used, would produce each possible outcome with the same probability? Can you describe such a procedure that does not rely on a computer or a calculator?

Let [math]X_1,\ X_2,\ \ldots,\ X_n[/math] be [math]n[/math] mutually independent random variables, each of which is uniformly distributed on the integers from 1 to [math]k[/math]. Let [math]Y[/math] denote the minimum of the [math]X_i[/math]'s. Find the distribution of [math]Y[/math].

A die is rolled until the first time [math]T[/math] that a six turns up.

• What is the probability distribution for [math]T[/math]?
• Find [math]P(T \gt 3)[/math].
• Find [math]P(T \gt 6 | T \gt 3)[/math].

If a coin is tossed a sequence of times, what is the probability that the first head will occur after the fifth toss, given that it has not occurred in the first two tosses?

A worker for the Department of Fish and Game is assigned the job of estimating the number of trout in a certain lake of modest size. She proceeds as follows: She catches 100 trout, tags each of them, and puts them back in the lake. One month later, she catches 100 more trout, and notes that 10 of them have tags.

• Without doing any fancy calculations, give a rough estimate of the number of trout in the lake.
• Let [math]N[/math] be the number of trout in the lake. Find an expression, in terms of [math]N[/math], for the probability that the worker would catch 10 tagged trout out of the 100 trout that she caught the second time.
• Find the value of [math]N[/math] which maximizes the expression in part (b). This value is called the maximum likelihood estimate for the unknown quantity [math]N[/math]. Hint: Consider the ratio of the expressions for successive values of [math]N[/math].

A census in the United States is an attempt to count everyone in the country. It is inevitable that many people are not counted. The U. S. Census Bureau proposed a way to estimate the number of people who were not counted by the latest census. Their proposal was as follows: In a given locality, let [math]N[/math] denote the actual number of people who live there. Assume that the census counted [math]n_1[/math] people living in this area. Now, another census was taken in the locality, and [math]n_2[/math] people were counted. In addition, [math]n_{12}[/math] people were counted both times.

• Given [math]N[/math], [math]n_1[/math], and [math]n_2[/math], let [math]X[/math] denote the number of people counted both times. Find the probability that [math]X = k[/math], where [math]k[/math] is a fixed positive integer between 0 and [math]n_2[/math].
• Now assume that [math]X = n_{12}[/math]. Find the value of [math]N[/math] which maximizes the expression in part (a). Hint: Consider the ratio of the expressions for successive values of [math]N[/math].