Question
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
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}
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