9A certain brand of copper wire has flaws about every 40 centimeters.

Model the locations of the flaws as a Poisson process. What is the probability

of two flaws in 1 meter of wire?

12.7 The Poisson model is sometimes used to study the flow of traffic ([15]).

If the traffic can flow freely, it behaves like a Poisson process. A 20-minute

time interval is divided into 10-second time slots. At a certain point along the

highway the number of passing cars is registered for each 10-second time slot.

Let nj be the number of slots in which j cars have passed for j = 0,..., 9.

Suppose that one finds

j 0 1 2 3 456789

nj 19 38 28 20 7 3 4 0 0 1

Note that the total number of cars passing in these 20 minutes is 230.

a. What would you choose for the intensity parameter ??

b. Suppose one estimates the probability of 0 cars passing in a 10-second

time slot by n0 divided by the total number of time slots. Does that

(reasonably) agree with the value that follows from your answer in a?

c. What would you take for the probability that 10 cars pass in a 10-second

time slot?

12.8 Let X be a Poisson random variable with parameter .

a. Compute E[X(X ? 1)].

b. Compute Var(X), using that Var(X) = E[X(X ? 1)] + E[X] ? (E[X])2.

12.9 Let Y1 and Y2 be independent Poisson random variables with parameter

1, respectively 2. Show that Y = Y1 + Y2 also has a Poisson distribution.

Instead of using the addition rule in Section 11.1 as in Exercise 11.2, you

can prove this without doing any computations by considering the number

of points of a Poisson process (with intensity 1) in two disjoint intervals of

length 1 and 2.

12.10 Let X be a random variable with a Pois () distribution. Show the

following. If

in k. If > 1, then the probabilities P(X = k) are first increasing, then

decreasing (cf. Figure 12.1). What happens if = 1?

12.11 Consider the one-dimensional Poisson process with intensity ?. Show

that the number of points in [0, t], given that the number of points in [0, 2t]

is equal to n, has a Bin(n, 1

2 ) distribution.

Hint: write the event {N([0, s]) = k, N([0, 2s]) = n} as the intersection of the

(independent!) events {N([0, s]) = k} and {N((s, 2s]) = n ? k}

In 10 Heads! 2. In Example 7.7, we observed a sample of size n = 100. A normal prob- ability plot and kernel density estimate constructed from this sample suggested that the observations had been drawn from a nonnormal dis- tribution. True or False: It follows from the Central Limit Theorem that a kernel density estimate constructed from a much larger sample would more closely resemble a normal distribution.Consider the triangle kernel, K(u), plotted below. 1.00 - 0.75 - K(u) 0.50 0.25 - 0.00 - -2 -1 N CO The triangle kernel is to be used as the basis for a kernel density estimate. As a first step in its construction, for each x in the data we construct , K (U Zi ). Suppose n = 3 and the data are x1 = 2, 12 = 4, x3 = 7 and use h = 0.3. Illustrate the result of this first step on the axes below.mean; median; mid-range; mode; quart ple variance; standard deviation; coefficient of outlier; inter-quartile range; symmetry; skewn tosis; normal distribution; box plot; histogra and leaf display; kernel density estimate; line chart; column chart; pie chart. 2.6 Exercises 1. Obtain Ezz for the following cases: (a) zi = 1 and n = 5. (b) zi = i and n = 5. (c) zi = 272 and n = 5. (d) zi = 1/i and n = 5