What is joint probability distribution

A joint probability distribution is a distribution of the probability of two or more events occurring simultaneously.

For example, the joint probability of two events $X$ and $Y$ is represented as $P(X,Y)$ or $P(X \cap Y)$, and the joint probability of three events $X$, $Y$ and $Z$ is represented as $P(X,Y,Z)$ or $P(X \cap Y \cap Z)$.

When the random variables are discrete, the distribution is a discrete joint probability distribution; when they are continuous, the distribution is a continuous joint probability distribution.

Discrete joint probability distribution

Suppose that the blood types of a certain elementary school class have the following distribution:

X\Y	Type A	Type B	Type O	Type AB
Boy	0.25	0.10	0.10	0.05
Girl	0.20	0.20	0.05	0.05

Considering event $X$ as boy or girl and event $Y$ as blood type, joint probability that event $X$ is a girl and event $Y$ is type O is 0.10.

Continuous joint probability distribution

The joint probability distribution of continuous random variables $X$ and $Y$ is expressed as follows:

$$P(a \leq X \leq b, c \leq Y \leq d) = \int^b_a \int^d_c f(x, y)dxdy$$

Since the sum of the probabilities is 1, the following equation holds

$$\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} f(x, y)dxdy = 1$$

As an example, consider the following probability density function:

$$f(x, y) = \left\{ \begin{array}{ll} x+y & (0 \leq x \leq 1, 0 \leq y \leq 1) \\ 0 & (otherwise) \end{array} \right.$$

The probability $P(0 \leq x \leq \frac{1}{2}, 0 \leq y \leq \frac{1}{2})$ with $0 \leq x \leq \frac{1}{2}$ and $0 \leq y \leq \frac{1}{2}$ can be obtained as follows.

$$\begin{aligned} P(0 \leq x \leq \frac{1}{2}, 0 \leq y \leq \frac{1}{2}) &= \int^{\frac{1}{2}}_{0} \int^{\frac{1}{2}}_{0} (x+y)dxdy \\ &= \int^{\frac{1}{2}}_{0}[\frac{1}{2}x^2 + yx]_0^{\frac{1}{2}} dy \\ &= \int^{\frac{1}{2}}_{0}(\frac{1}{8} + \frac{1}{2}y)dy \\ &= [\frac{1}{8}y + \frac{1}{4}y^2]^{\frac{1}{2}}_0 \\ &= \frac{1}{8} \end{aligned}$$