2.1.1 Binomial Distribution

A Binomial distribution is a collect of trials where each trial has only two possible outcomes, success or failure. The key properties for a Binomial distribution are: a trial is repeated $n$ times, where $n$ is fixed; the probabilities of success and failure are constant across trials; and each trial is independent of the others. We characterize the distribution by $n$, the number of trials and $\theta$ the probability of success. 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 = $n\theta$.

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  Standard deviation = $n\theta (1-\theta )$.

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

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  PMF: $p(r)=\mathbb{P}(R=r)=\left(\stackrel{n}{r}\right){\theta }^r{(1-\theta )}^{n-r}$.