Probability Distributions
A probability distribution describes how the values of a random variable are distributed. It gives the probability that a random variable will take on each of its possible values.
Types of Probability Distributions
Discrete Probability Distributions
Include probability mass functions (PMFs) which assign probabilities to discrete outcomes, such that:
$\begin{gather} 0 \le \mathbb{P}(x) \le 1 \qquad \forall x \\ \mathbb{P}(x=y)=\mathbb{P}(y) \qquad \forall y \\ \sum_X \mathbb{P}(x) = 1 \end{gather}$For example:

Bernoulli Distribution: The Bernoulli distribution describes experiments that have binary outcomes, e.g., result in either a success or a failure.

Example: Flipping a coin once, where heads is considered a success.

Formula:
$\mathbb{P}(x = k) = p^k (1p)^{1k} \quad \forall k \in \{0,1\}$Where $p$ is the probability of success, $1p$ is the probability of failure, and $k$ can be either 0 (failure) or 1 (success).


Poisson Distribution: The Poisson distribution describes the number of events occurring within a fixed interval of time or space when these events happen independently of each other and at a constant rate.

Example: The number of emails you receive in an hour.

Formula:
$\mathbb{P}(x = k) = \frac{\lambda^k e^{\lambda}}{k!}$Where $\lambda$ is the average number of events in the interval, $k$ is the number of occurrences, and $e$ is the Euler's constant.

Continuous Probability Distributions
Include probability density functions (PDFs) which describe the likelihood of a continuous random variable falling within a particular range of values, such that:
$\begin{gather} 0 \le f(x) \\ \mathbb{P}(a \le x \le b) = \int_a^b f(x) \,dx \\ \int_\infty^{\infty} f(x) \,dx = 1 \end{gather}$For example:

Uniform Distribution: The uniform distribution describes an equal probability for all values within a specified range.

Example: The probability of randomly selecting a number between 1 and 10.

Formula:
$f(x) = \begin{cases} \frac{1}{ba} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$Where $a$ and $b$ are the minimum and maximum values of the range.


Normal Distribution: The normal distribution, often referred to as the Gaussian distribution or bell curve, describes data that clusters around a mean.

Example: Heights of people in a population.

Formula:
$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{(x\mu)^2}{2\sigma^2}}$Where $\mu$ is the mean and $\sigma$ is the standard deviation.


Student's tDistribution: The Student's tdistribution is used to estimate population parameters when the sample size is small and the population standard deviation is unknown.

Example: The distribution of the sample mean for small sample sizes.

Formula:
$f(x) = \frac{\Gamma \left(\frac{\nu+1}{2}\right)}{\sqrt{\nu \pi} \Gamma \left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{\frac{\nu+1}{2}}$Where $\nu$ is the degrees of freedom, and $\Gamma$ the gamma function.
