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I was watching the harvard stat110 course on Geometric distribution and the PMF is $P(X = k) = pq^k$ where $p$ is the probability of success and $q = 1-p$. later he mentions another r.v $Y$ which he defined as the 1st success distribution and he mentions that this distribution, unlike the Geometric Distribution, includes the first success.

I got a bit confused as to the meaning of including the first success, in the PMF of $Geo(p)$ there is a $p$, isn't this including the first success already?

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There are two common forms of the geometric distribution.

Imagine we have a sequence of Bernoulli trials (specifically, imagine we're tossing a biased coin).

I'm interested in $Y$, the number of trials to get the first head (how many tosses did I make to get a head), so let's call "got a head" a success, and it's the number of trials to get the first success. If I toss T, T, H, I count 3 trials.

My sister is interested in the number of times I failed to get a head before I succeeded, $X$ (i.e. she only cares how many tails I got). If I toss T, T, H she counts "two failures".

Both these variables are called "geometric" and the only difference is that my variable counts the trial on which I got a success in the total while my sisters variable does not (i.e. $Y=X+1$)

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  • $\begingroup$ +1: Crystal clear $\endgroup$ Oct 4, 2023 at 11:22

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