The Geometric distribution is used when we want to find the probability of performing a sequence of
trials with X “failures” before the first “success”, where a “success” occurs with some probability p.
We can denote such a random variable as X ∼ Geo(p). The pmf is given by
f(x) = p(1 − p)^x, x = 0,1,2, . . .
where 0 < p < 1.
(a) Is X a discrete or continuous random variable?
(b) Show that f(x) is a valid pmf.
(c) In a series of coin flips, what is the probability of getting the first heads on the 5th flip (so that
there are 4 tails before the first heads)?
I know it is discrete, and I know for part (b) that it has to be non negative and the integral equal to 1. However, when I work out the integral I get an awkward log function. And part (c), the only way I can think of is doing (1/2)^5, since each flip is independent but I feel like I am missing something.
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