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Chapter 6: Q.6.21 (page 276)

Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters (t, β). That is, its density is f(w) = βe−βw(βw)t−1/(t), w > 0. Suppose also that given that W = w, the number of accidents during that day—call it N—has a Poisson distribution with mean w. Show that the conditional distribution of W given that N = n is the gamma distribution with parameters (t + n, β + 1)

Short Answer

Expert verified

In order to obtain the required conditional distribution, use the definition of conditional PDF.

Step by step solution

01

Content Introduction

A random variable is a variable with an unknown value or a function that gives values to each of the results of an experiment. It's possible for a random variable to be discrete or continuous.

02

Content Explanation

We are required to find the distribution of W given that N = n. We have that,

fW,N(w,n)=fW,N(w,n)P(N=n)=P(N=n|W=w)fW(w)P(N=n)

Since, we have that N/W=w~Pois(w)we have that

P(N=n,W=w)=wnn!e-w

Thus,

P(N=n,W=w)fW(w)P(N=n)=wnn!e-w.βe-βw(βw)t-1τ(t)P(N=n)

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The gross weekly sales at a certain restaurant are a normal random variable with mean\(2200and standard deviation \)230. What is the probability that

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