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Consider a sequence of independent trials of an experiment where each trial can result in one...

Consider a sequence of independent trials of an experiment where each trial can result in one of two possible outcomes, “Success” or “Failure”. Suppose that the probability of success on any one trial is p. Let X be the number of trials until the rth success is observed, where r ≥ 1 is an integer.

(a) Derive the probability mass function (pmf) for X. Show your work.

(b) Name the distribution by matching your resulting pmf up with one in the table of distributions.

(c) Find the distribution of Y = X−r. (Name it!) What is the interpretation of Y in terms of the success/failure experiment?

(d) Give an alternative name for the distribution from part (b) in the case when r = 1.

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