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4.9: Formula Review

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    4.1 Hypergeometric Distribution

    \[
    h(x)=\frac{\binom{A}{x}\binom{N-A}{n-x}}{\binom{N}{n}}
    \]

    4.2 Binomial Distribution

    \(X \sim B(n, p)\) means that the discrete random variable \(X\) has a binomial probability distribution with \(n\) trials and probability of success \(p\).
    \(X=\) the number of successes in \(n\) independent trials
    \(n=\) the number of independent trials
    \(X\) takes on the values \(x=0,1,2,3, \ldots, n\)
    \(p=\) the probability of a success for any trial
    \(q=\) the probability of a failure for any trial
    \[
    \begin{array}{l}
    p+q=1 \\
    q=1-p
    \end{array}
    \]

    The mean of \(X\) is \(\mu=n p\). The standard deviation of \(X\) is \(\sigma=\sqrt{n p q}\).
    \[
    P(x)=\frac{n!}{x!(n-x)!} \cdot p^x q^{(n-x)}
    \]
    where \(P(X)\) is the probability of \(X\) successes in \(n\) trials when the probability of a success in ANY ONE TRIAL is \(p\).

    4.3 Geometric Distribution

    \[
    P(X=x)=p(1-p)^{x-1}
    \]
    \(X \sim \mathrm{G}(\mathrm{p})\) means that the discrete random variable \(X\) has a geometric probability distribution with probability of success in a single trial \(p\).
    \(X=\) the number of independent trials until the first success
    \(X\) takes on the values \(x=1,2,3, \ldots\)
    \(p=\) the probability of a success for any trial
    \(q=\) the probability of a failure for any trial \(p+q=1\)
    \(q=1-p\)
    The mean is \(\mu=\frac{1}{p}\).
    The standard deviation is \(\sigma=\sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\).

    4.4 Poisson Distribution

    \(X \sim P(\mu)\) means that \(X\) has a Poisson probability distribution where \(X=\) the number of occurrences in the interval of interest.
    \(X\) takes on the values \(x=0,1,2,3, \ldots\)
    The mean \(\mu\) or \(\lambda\) is typically given.
    The variance is \(\sigma^2=\mu\), and the standard deviation is
    \[
    \sigma=\sqrt{\mu}
    \]

    When \(P(\mu)\) is used to approximate a binomial distribution, \(\mu=n p\) where \(n\) represents the number of independent trials and \(p\) represents the probability of success in a single trial.
    \[
    P(x)=\frac{\mu^x e^{-\mu}}{x!}
    \]

    4.9: Formula Review is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.