4.5: Chapter Formula Review

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Hypergeometric Distribution

\(h(x)=\frac{\left(\begin{array}{l}{A} \\ {x}\end{array}\right)\left(\begin{array}{l}{N-A} \\ {n-x}\end{array}\right)}{\left(\begin{array}{l}{N} \\ {n}\end{array}\right)}\)

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 \(x = 0, 1, 2, 3, ..., n\)

\(p =\) 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 of \(X\) is \(\mu = np\). The standard deviation of \(X\) is \(\sigma=\sqrt{n p q}\).

\[P(x)=\frac{n !}{x !(n-x) !} \cdot p^{x} q^{(n-x)}\nonumber\]

where \(P(X)\) is the probability of \(X\) successes in \(n\) trials when the probability of a success in ANY ONE TRIAL is \(p\).

Geometric Distribution

\(P(X=x)=p(1-p)^{x-1}\)

\(X \sim G(p)\) 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 \(x = 1, 2, 3, ...\)

\(p =\) 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 \(\sigma=\sqrt{\frac{1-p}{p^{2}}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\).

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 \(x = 0, 1, 2, 3, ...\)

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, \(\mu = np\) where n represents the number of independent trials and \(p\) represents the probability of success in a single trial.

\[P(x)=\frac{\mu^{x} e^{-\mu}}{x !}\nonumber\]