# Thread: y follows a Poisson distribution with rate lambda*x

1. ## y follows a Poisson distribution with rate lambda*x

Assume that the number of faults in the roll, say y, follows a Poisson distribution with rate lambda*x, where x is the length of the roll.
Does this mean the pmf is
f(y; lambda*x) = exp(-lambda*x)(lambda*x)^y / y! ?

Thanks

2. ## Re: y follows a Poisson distribution with rate lambda*x

Yes, you are correct

3. ## Re: y follows a Poisson distribution with rate lambda*x

Thanks!!

So if lambda follows a Gamma distribution with shape and scale parameters alpha and beta, how do I go about finding the distribution of y (which follows the Poisson distribution with rate lamba*x) ?

Thanks again!

4. ## Re: y follows a Poisson distribution with rate lambda*x

Given

Then

The conditional p.m.f. and the density is already known.
And you just need to evaluate this integral.

5. ## Re: y follows a Poisson distribution with rate lambda*x

So I have
and

Now I need to find
Should I have

or should I have

Which one makes sense??

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