Probability Distributions

Discrete random variables

Expected value

E (X) = μ_{X} = \sum x_{i} P (x_{i})

Variance

σ_{X}^{2} = \sum (x_{i} - μ_{X})^{2} P (x_{i}) = E (X^{2}) - [E (X)]^{2}

Linear change, Y = a + bX

μ_{Y} = a + b μ_{X} σ_{Y} = | b | σ_{X}

Sum or difference of independent X and Y

μ_{X \pm Y} = μ_{X} \pm μ_{Y} σ_{X \pm Y}^{2} = σ_{X}^{2} + σ_{Y}^{2}

Probability

P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}

Mean and standard deviation

μ = n p σ = \sqrt{n p (1 - p)}

Probability

P (X = k) = (1 - p)^{k - 1} p

Mean and standard deviation

μ = \frac{1}{p} σ = \frac{\sqrt{1 - p}}{p}

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