What is Probability Distribution?
Probability distribution is a function that describes the probability of a random variable taking certain values.
Types of Probability Distribution
Discrete Probability Distribution
Continuous Probability Distribution
Probability Distribution Functions
\[f(x;a,b) = \left\{ \begin{array}{ll} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{array} \right.\]
Normal Distribution
\[f(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
Bernoulli Distribution
\[f(x;p) = \left\{ \begin{array}{ll} p & x = 1 \\ 1-p & x = 0 \end{array} \right.\]
Binomial Distribution
\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\] \[f(x;n,p) = \binom{n}{x} p^x (1-p)^{n-x}\]
Poisson Distribution
\[f(x;\lambda) = \frac{\lambda^x e^{-\lambda}}{x!}\]
Exponential Distribution
\[f(x;\lambda) = \left\{ \begin{array}{ll} \lambda e^{-\lambda x} & x \geq 0 \\ 0 & x < 0 \end{array} \right.\] \[F(x) = \left\{ \begin{array}{ll} 1 - e^{-\lambda x} & x \geq 0 \\ 0 & x < 0 \end{array} \right.\]
Gamma Distribution
\[\begin{aligned} \Gamma(\alpha) &= \int_0^\infty x^{\alpha-1} e^{-x} dx\\ \Gamma(\alpha + 1) &= \int_0^\infty x^{\alpha} e^{-x} dx\\ &= \int_0^\infty x^{\alpha} d(-e^{-x})\\ &= \left. x^{\alpha} (-e^{-x}) \right|_0^\infty - \int_0^\infty \alpha x^{\alpha-1} (-e^{-x}) dx\\ &= 0 + \alpha \int_0^\infty x^{\alpha-1} e^{-x} dx\\ &= \alpha \Gamma(\alpha)\\ \Gamma(n+1) &= n!\\ \end{aligned}\] \[f(x;\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}\]
Beta Distribution
\[f(x;\alpha,\beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}\]
Rayleigh Distribution
\[f(x;\sigma) = \frac{x}{\sigma^2} e^{-\frac{x^2}{2\sigma^2}}\] \[F(x) = 1 - e^{-\frac{x^2}{2\sigma^2}}\]
Weibull Distribution
\[f(x;\lambda,k) = \left\{ \begin{array}{ll} \frac{k}{\lambda} (\frac{x}{\lambda})^{k-1} e^{-(\frac{x}{\lambda})^k} & x \geq 0 \\ 0 & x < 0 \end{array} \right.\] \[F(x) = 1 - e^{-(\frac{x}{\lambda})^k}\]