Statistical Distributions Overview

• Provides an overview of statistical distributions
• Details both discrete and continuous distributions

Probability Distributions

• Tables presented for discrete and continuous distributions, outlining key properties

Discrete Distributions

• Bernoulli distribution models the probability of success or failure of a single trial
  • pmf: probability mass function is $p^x(1-p)^{1-x}$
  • Mean: $p$
  • Variance: $p(1-p)$
  • Support: $x \in {0, 1}$
• Binomial distribution counts the number of successes in a fixed number of independent Bernoulli trials
  • pmf: ${n \choose x}p^x(1-p)^{n-x}$
  • Mean: $np$
  • Variance: $np(1-p)$
  • Support: $x \in {0, 1, 2,..., n}$
• Geometric distribution models the number of trials needed to get the first success in a series of Bernoulli trials
  • pmf: $p(1-p)^{x-1}$
  • Mean: $\frac{1}{p}$
  • Variance: $\frac{1-p}{p^2}$
  • Support: $x \in {1, 2, 3,...}$
• Negative Binomial distribution models the number of trials needed to achieve a fixed number of successes
  • pmf: ${x-1 \choose r-1}p^r(1-p)^{x-r}$
  • Mean: $\frac{r}{p}$
  • Variance: $\frac{r(1-p)}{p^2}$
  • Support: $x \in {r, r+1, r+2,...}$
• Hypergeometric distribution models the probability of successes in a sample without replacement
  • pmf: $\frac{{K \choose x}{N-K \choose n-x}}{{N \choose n}}$
  • Mean: $\frac{nK}{N}$
  • Variance: $n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}$
  • Support: $max(0, n-(N-K)) \le x \le min(n, K)$
• Poisson distribution models the number of events occurring in a fixed interval of time or space
  • pmf: $\frac{\lambda^xe^{-\lambda}}{x!}$
  • Mean: $\lambda$
  • Variance: $\lambda$
  • Support: $x \in {0, 1, 2,...}$
• Discrete Uniform distribution assigns equal probability to each outcome in a finite set
  • pmf: $\frac{1}{b-a+1}$
  • Mean: $\frac{a+b}{2}$
  • Variance: $\frac{(b-a+1)^2 - 1}{12}$
  • Support: $x \in {a, a+1, a+2,..., b}$

Continuous Distributions

• Uniform distribution assigns constant probability density over an interval
  • pdf: $\frac{1}{b-a}$
  • Mean: $\frac{a+b}{2}$
  • Variance: $\frac{(b-a)^2}{12}$
  • Support: $x \in [a, b]$
• Exponential distribution models the time until an event occurs in a Poisson process
  • pdf: $\lambda e^{-\lambda x}$
  • Mean: $\frac{1}{\lambda}$
  • Variance: $\frac{1}{\lambda^2}$
  • Support: $x \in [0, \infty)$
• Gamma distribution generalizes the exponential distribution and models the time until multiple events occur in a Poisson process
  • pdf: $\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$
  • Mean: $\frac{\alpha}{\beta}$
  • Variance: $\frac{\alpha}{\beta^2}$
  • Support: $x \in [0, \infty)$
• Chi-squared distribution is the distribution of a sum of squared standard normal variables
  • pdf: $\frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2 - 1}e^{-x/2}$
  • Mean: $k$
  • Variance: $2k$
  • Support: $x \in [0, \infty)$
• Normal distribution is a bell-shaped distribution often used to model real-valued random variables with unknown distributions
  • pdf: $\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$
  • Mean: $\mu$
  • Variance: $\sigma^2$
  • Support: $x \in (-\infty, \infty)$
• Bivariate Normal distribution extends the normal distribution to model two correlated variables
  • Means: $\mu_x, \mu_y$
  • Variances: $\sigma_x^2, \sigma_y^2$
  • Support: $x, y \in (-\infty, \infty)$

Notes

• pmf refers to the probability mass function
• pdf refers to the probability density function