Basic Probability Nomenclature

probability bayesTheorem setTheory

Probability Density Function

Let $p(x)$ be a random variable $x$. The probability density function can be defined as.

$${\int }_a^{b}p(x)dx$$

Which tells us

$$p(a\le x\le b)$$

The whole distribution adds to 1

$${\int }_{-\infty }^{\infty }p(x)dx=1$$

Multivariate Representation

This is if we have a probability density function in multiple dimensions.

$${\int }_{\mathbf{a}}^{\mathbf{b}}p(\mathbf{x})d\mathbf{x}=\int \int \int \int p(x_1,x_2,x_3,..,x_N)d{x}_{1}d{x}_{2}d{x}_3\dots d{x}_N$$

Cumulative Density Function

$${\int }_{-\infty }^{b}p(x)dx=p(x\le b)$$

Joint probability

$$p(\mathbf{x},\mathbf{y})=p(\mathbf{x}|\mathbf{y})p(\mathbf{y})=p(\mathbf{y}|\mathbf{x})p(\mathbf{x})$$

This gives you Bayes’ Theorem