Question

In: Statistics and Probability

Alex met Claire Boucher (Grimes) at McGill. He has been waiting for the release of her...

Alex met Claire Boucher (Grimes) at McGill. He has been waiting for the release of her next album. Assume that the waiting time is exponential with mean 3 years. To keep up with releases Alex receives Resident Advisor’s monthly album review newsletter. Assume that the album will be featured in the next issue after its release. Let X be the number of newsletters required to get news of the release of the album. Find the probability mass function of X.

Expert Solution

Let X be the waiting time for release of the album which follows exponential distribution with pdf

where =1/3 as mean is given 3 years.

And the cdf of X is

Let Y be the number of newsletters required to get news of release of the album. Then Y takes values form {1,2,3,...}.

Then the relationship between waiting time X and Y is given by the following equation

where denotes the greatest integer less than or equal to X. The relationship comes from the fact that the album will be featured in the next issue of the newsletter after its release.

Then the probability mass function of Y is given by

; where k is observed valued of Y.

Then pmf of Y can be written as

where is the probability that Alex gets the news of release of the album.

So the pmf of Y is given by

which is Geometric distribution with .

Then it follows Geometric distribution with

orchestra answered 2 years ago

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