Probability Distribution

| Tags Math

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

$$f(x \mid \theta) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$$
• $$\beta > 0$$$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

$$\chi^2_n = X_1^2 + X_2^2 + … + X_n^2$$
• $$Y \sim N(0, 1)$$$F Distribution $$F = \frac{\frac{X_1^2 + X_2^2 + … + X_n^2}{n}}{\frac{Y_1^2 + Y_2^2 + … + Y_m^2}{m}}$$ • $$X_i \sim N(0, 1)$$$
• $$Y_i \sim N(0, 1)$$\$