Basic reference on distributions used in statistics.
Discrete distributions
Bernoulli distribution
Single trial whose outcome can be success or failure.
\[ P(X = 1) = p \] \[ P(X = 0) = 1-p \] \[ \mathrm{E}[X] = p \] \[ \mathrm{Var}[X] = p(1-p) \]
Geometric distribution
Number of trials until first success.
For \(n = 1, 2, \ldots\),
\[ P(X = n) = (1-p)^{n-1}p \] \[ \mathrm{E}[X] = \frac{1-p}{p} \] \[ \mathrm{Var}[X] = \frac{1-p}{p^2} \]
Binomial distribution
Number of successes in \(n\) trials.
For \(0 \leq k \leq n\),
\[ P(X = k) = {n \choose k} p^x (1-p)^{n-x} \] \[ \mathrm{E}[X] = np \] \[ \mathrm{Var}[X] = np(1-p) \]
Poisson distribution
Distribution of rare events in a large population.
For \(n = 0, 1, 2, \ldots\),
\[ P(X = n) = \frac{e^{-\lambda} \lambda^n}{n!} \] \[ \mathrm{E}[X] = \lambda \] \[ \mathrm{Var}[X] = \lambda \]
Continuous distributions
Uniform distribution
For \(a \leq x \leq b\),
\[ f(x) = \frac{1}{b-a} \] \[ \mathrm{E}[X] = \frac{a+b}{2} \] \[ \mathrm{Var}[X] = \frac{(b-a)^2}{12} \]
Exponential distribution
For \(x \geq 0\),
\[ f(x) = \lambda e^{-\lambda x} \] \[ \mathrm{E}[X] = \frac{1}{\lambda} \] \[ \mathrm{Var}[X] = \frac{1}{\lambda^2} \]
Normal distribution
\[ f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2 \sigma^2} (x - \mu)^2} \] \[ \mathrm{E}[X] = \mu \] \[ \mathrm{Var}[X] = \sigma^2 \]
Gamma distribution
For \(x \geq 0\),
\[ f(x) = \frac{1}{\Gamma(\alpha)} \beta^\alpha x^{\alpha-1} e^{-\beta x} \] \[ \mathrm{E}[X] = \frac{\alpha}{\beta} \] \[ \mathrm{Var}[X] = \frac{\alpha}{\beta^2} \] \[ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x} \, dx \]
Properties of the Gamma function
For \(\alpha = 1\),
\[ \Gamma(1) = 1 \]
For \(\alpha > 1\),
\[ \Gamma(\alpha) = (\alpha - 1) \Gamma(\alpha - 1) \]
For integer \(n \geq 1\),
\[ \Gamma(n) = (n-1)! \]