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