2.1.2 Geometric Distribution

A Geometric distribution can answer two questions. Either a) the probability of getting first success on the $n$th trial, or b) the probability of getting success on $\le n$ trials. As for the Binomial distribution the attempts are independent and the probability of success, $\theta$ is constant across attempts. Key quantities are:

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  $\theta$ = probability of success.

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  $1-\theta$ = probability of failure.

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  Mean = $\frac{1-\theta }{\theta }$.

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  Variance = $\frac{1-\theta }{{\theta }^2}$.

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  Note that $\theta \in (0,1)$ as it is a probability.

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  PMF: $p(r)=\mathbb{P}(R=r)=\theta {(1-\theta )}^r$, Probability of first having success on the $(r+1)$th trial.