In one of the first studies of the Poisson distribution, von Bortkiewicz^{[Notes 1]} considered the frequency of deaths from kicks in the Prussian army corps. From the study of 14 corps over a 20-year period, he obtained the data shown in Table.

Fit a Poisson distribution to this data and see if you think that the Poisson distribution is appropriate.

Notes

It is often assumed that the auto traffic that arrives at the intersection during a unit time period has a Poisson distribution with expected value [math]m[/math]. Assume that the number of cars [math]X[/math] that arrive at an intersection from the north in unit time has a Poisson distribution with parameter [math]\lambda = m[/math] and the number [math]Y[/math] that arrive from the west in unit time has a Poisson distribution with parameter [math]\lambda = \bar m[/math]. If [math]X[/math] and [math]Y[/math] are independent, show that the total number [math]X + Y[/math] that arrive at the intersection in unit time has a Poisson distribution with parameter [math]\lambda = m + \bar m[/math].

Cars coming along Magnolia Street come to a fork in the road and have to choose either Willow Street or Main Street to continue. Assume that the number of cars that arrive at the fork in unit time has a Poisson distribution with parameter [math]\lambda = 4[/math]. A car arriving at the fork chooses Main Street with probability 3/4 and Willow Street with probability 1/4. Let [math]X[/math] be the random variable which counts the number of cars that, in a given unit of time, pass by Joe's Barber Shop on Main Street. What is the distribution of [math]X[/math]?

In the appeal of the People v. Collins case (see Exercise), the counsel for the defense argued as follows: Suppose, for example, there are 5,00,00 couples in the Los Angeles area and the probability that a randomly chosen couple fits the witnesses' description is 1/12,00,00. Then the probability that there are two such couples given that there is at least one is not at all small. Find this probability. (The California Supreme Court overturned the initial guilty verdict.)

A manufactured lot of brass turnbuckles has [math]S[/math] items of which [math]D[/math] are defective. A sample of [math]s[/math] items is drawn without replacement. Let [math]X[/math] be a random variable that gives the number of defective items in the sample. Let [math]p(d) = P(X = d)[/math].

• Show that
  [[math]] p(d) = \frac{{D \choose d} {{S - D} \choose {s - d}}}{{S \choose s}}\ . [[/math]]
  Thus, X is hypergeometric.
• Prove the following identity, known as Euler's formula:
  [[math]] \sum_{d = 0}^{\min(D,s)}{ D \choose d} {{S - D} \choose {s - d}} = {S \choose s}\ . [[/math]]

A bin of 1000 turnbuckles has an unknown number [math]D[/math] of defectives. A sample of 100 turnbuckles has 2 defectives. The maximum likelihood estimate for [math]D[/math] is the number of defectives which gives the highest probability for obtaining the number of defectives observed in the sample. Guess this number [math]D[/math] and then write a computer program to verify your guess.

There are an unknown number of moose on Isle Royale (a National Park in Lake Superior). To estimate the number of moose, 50 moose are captured and tagged. Six months later 200 moose are captured and it is found that 8 of these were tagged. Estimate the number of moose on Isle Royale from these data, and then verify your guess by computer program (see Exercise Exercise).

A manufactured lot of buggy whips has 20 items, of which 5 are defective. A random sample of 5 items is chosen to be inspected. Find the probability that the sample contains exactly one defective item

• if the sampling is done with replacement.
• if the sampling is done without replacement.

Suppose that [math]N[/math] and [math]k[/math] tend to [math]\infty[/math] in such a way that [math]k/N[/math] remains fixed. Show that

A bridge deck has 52 cards with 13 cards in each of four suits: spades, hearts, diamonds, and clubs. A hand of 13 cards is dealt from a shuffled deck. Find the probability that the hand has

• a distribution of suits 4, 4, 3, 2 (for example, four spades, four hearts, three diamonds, two clubs).
• a distribution of suits 5, 3, 3, 2.