Which definition is correct for a geometric random variable?

Question

Is it

• The number of failures BEFORE the first success OR
• The number of trials required to get a first success?

Also, if I was to work out the expected value of a geometric random variable, say $p = 0.25$ (Expected value = $3$), does that mean that I will have $3$ failures AND THEN a success, or $2$ failures and then a success??

I would immensely appreciate some help here. Thank you so much x

Answer 1

I would say both are correct, just try to be consistent and use only one. However, I found the one with the number of trials required more appealing, since $E[X]$ is simply $1/p$ in that case.

PS: if you are given both $p = 0.25$ and $E[X] = 3$, then you can check, which definition is used: if this was the number-of-trials definition, you would get $$E[X] = 1/p = 1/0.25 = 4 \neq 3\text{,}$$ hence the number-of-failures definition was used.

Answer 2

By way of clarification,

if $X$ is the (discrete r.v. denoting) number of failures, $Y$ likewise is the (discrete r.v. denoting) number of (independent Bernoulli) trials [ including the $F$(irst)$S$(uccess) ], we always have $Y= X+1$ and we say

$X$ ~ $Geom(p)$ and $Y$ ~ $FS(p)$ where $p$ is the success probability of each of the trials and $E(X) = (1-p)/p$ and $E(Y) = E(X+1) = E(X)+1 = 1/p$.

[ cf Joseph Blitzstein et al "Intro. to P - CRC Press 2015" for excellent discussion, story proofs etc.. ]