A hurricane just blew across the coast and flung a school of fish onto the road nearby the beach.

The road starts at your house and is infinitely long. We will label a point on the road by its distance

from your house (in miles). For each n ∈ N, the number of fish that land on the segment of the road

[n,n + 1] is independently pois(λ) and each fish that is flung into that segment of the road lands

uniformly at random within the segment. Keep in mind that you can cite any result from lecture or

discussion without proof.

(a) What is the distribution of the number of fish arriving in segment [0,n] of the road, for some

n ∈ N.

(b) Let [a,b] be an interval in [0,1]. What is the distribution of the number of fish that lands in the

segment [a,b] of the road?

(c) Let [a,b] be any interval such that a ≥ 0. What is the distribution of the number of fish that

land in [a,b]?

(d) Suppose you take a stroll down the road. What is the distribution of the distance you walk (in

miles) until you encounter the first fish? Justify with proof.

(e) Suppose you encounter a fish at distance x. What is the distribution of the distance you walk

until you encounter the next fish?

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