A listing of all the values the random variable can take with their corresponding probabilities make a probability distribution. It can be classified into two categories, namely Discrete Probability Distribution and Continuous Probability Distribution.
Discrete Probability Distribution
In Discrete Probability Distribution, the random variable can only be assigned with a discrete value which is finite or countable, common discrete probability distributions are:
Discrete uniform distribution
$$ P(k \mid m, n) = \frac{1}{n-m-1}$$Bernoulli(0-1) distribution
$$ P(1 \mid p) = p $$Binomial distribution
$$ P(k \mid n, p) = \lgroup ^n_k \rgroup p^k (1-p)^{n-k} $$Multinomial distribution
$$ P(\vec{x} \mid n, \vec{p}) = \frac{n!}{x_1!x_2!...x_k!} p_1^{x_1}p_2^{x_2}…p_k^{x_k} $$Geometric distribution
$$ P(k \mid p) = (1-p)^kp$$Hypergeometric distribution
$$ P(k \mid N, K, n) = \frac{(^K_k)(^{N-K}_{n-k})}{(^N_n)} $$Poisson distribution
$$ P(k \mid \lambda) = \frac{\lambda^k}{k!} e^{-\lambda} $$Continuous Probability Distribution
In Continuous Probability Distribution, the random variable can take a continuous range of values, common continuous probability distributions are:
Uniform Distribution
$$ f(x \mid a, b) = \frac{1}{b-a}$$Normal(Gaussian) Distribution
$$ f(x \mid \mu , \sigma) = \frac{1}{\sigma \sqrt{2\pi}}e^{- \frac{(x-\mu)^2}{2\sigma^2}}$$Exponential Distribution
$$ f(x \mid \theta) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$$- $$$ \theta > 0 $$$, and it is the expect of this distribution.
Gamma Distribution
$$ f(x \mid \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} $$- $$$\alpha > 0$$$, and it is shape parameter
- $$$\beta > 0$$$, and it is rate parameter
Beta Distribution
$$ f(x \mid \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)}x^{\alpha-1} (1-x)^{\beta-1}$$- $$$x$$$ is the random variable, and s.t. $$$ 0 < x < 1 $$$
- $$$ \alpha > 0 $$$
- $$$ \beta > 0 $$$
Dirichlet Distribution
$$ Dir(\vec{p} \mid \vec{\alpha}) = \frac{\Gamma(\sum^K_1 \alpha_k)}{\prod^K_1 \Gamma(\alpha_k)} \prod^K_1 p_k^{\alpha_k-1}$$$$$\chi^2$$$ Distribution
$$ \chi^2_n = X_1^2 + X_2^2 + … + X_n^2 $$- $$$ X_i \sim N(0, 1) $$$
Student's T Distribution
$$ t = \frac{Y}{\sqrt{\frac{X_1^2 + X_2^2 + … + X_n^2}{n}}}$$- $$$ X_i \sim N(0, 1) $$$
- $$$ Y \sim N(0, 1) $$$
F Distribution
$$ F = \frac{\frac{X_1^2 + X_2^2 + … + X_n^2}{n}}{\frac{Y_1^2 + Y_2^2 + … + Y_m^2}{m}} $$- $$$ X_i \sim N(0, 1) $$$
- $$$ Y_i \sim N(0, 1) $$$