Manjeet Dahiya

Joint Probability Distribution

The notion of probability distribution of a random variable can be extended to the joint distribution of multiple random variables. On the same lines, we can define the joint pdf, joint cumulative distribution and other various definitions for multiple random variables.

Bivariate/Multivariate Distributions

The joint distribution of two random variables $X$ and $Y$ is the collection of probabilities of the form $Pr((X, Y) \in C)$ for all sets $C$ of pairs of real numbers such that ${(X, Y) \in C}$ is an event.

If the values taken by $(X, Y)$ are finite or countable, then the distribution is called discrete joint distribution. If the values taken by $(X, Y)$ are un-countably infinite, then the distribution is called continuous joint distribution.

Similar to the single random variable case, the joint distribution is defined in terms of pf or pdf.

A joint distribution is called bivariate if the number of random variables involved is two and multivariate for more than two.

Another way to interpret joint distribution for discrete random variables $X$ and $Y$ is a table whose number of entries is the product of the number of entries in the distribution of $X$ and $Y$. In the vertical direction, random variable $X$ takes different values, and in the horizontal direction, random variable $Y$ takes different values. For every combination of $X$ and $Y$, the table lists the probability of the event $X=x \land Y=y$.

Cumulative distributions

Cumulative distribution function for joint distribution is defined as:

$F(x, y) = Pr(X \le x\ and\ Y \le y)$

Marginal Distribution

Given a joint distribution we can find the distribution of a single random variable from it. The distribution of a random variable computed from a joint distribution is called marginal distribution. Consequently, we can define marginal pf/pdf as well as marginal cdf.

The term marginal comes from the fact that these probabilities occur at the margins of the table in case of discrete joint distributions.

Marginal pf/pdf

$f_X(x) = \int_{-\infty}^{\infty} f(x, y) dy$, similarly for Y.

$f_X(x) = \sum \limits_{y \in domain(Y)} f(x, y)$

Note the marginal pf/pdf is a valid probability distribution, hence, all the rules of probability distribution apply to it.

Marginal cdf

$F_X(x) = lim_{y \rightarrow \infty} F(x, y)$ similarly for Y.

Independence of random variables

Two random variable $X$ and $Y$ and independent if for every events $A$ and $B$ the following holds:

\[Pr(X \in A \land Y \in B) = Pr(X \in A) Pr(Y \in B)\]

Conditional distributions

Conditional distribution is a generalization of conditional probability. Conditional distribution represents collection of probabilities when an event of one random variable is conditional of an event of the other random variable.

Probability of event $x$ of $X$ conditioned on event $y$ of $Y$ is written as:

\[Pr(X = x | Y = y) = Pr(X = x \land Y = y)/ Pr(Y = y) = f(x, y)/f_Y(y)\]

Here $f(x, y)$ is the joint pf/pdf and $f_Y(y)$ is the marginal pf/pdf of $Y$.

The equation is commonly written as follows ($\land$ is replaced by comma, and events have been omitted).

\[Pr(X | Y) = Pr(X, Y)/ Pr(Y)\]

equivalent to

\[Pr(X, Y) = Pr(X | Y) Pr(Y)\]

This is also called the chain rule of probability for random variables. For three random variables, it is written as:

\[Pr(X_1, X_2, X_3) = Pr(X_1 | X_2, X_3) Pr(X_2, X_3)\]

Conditional distribution

Conditional distribution of random variable $Y$ conditioned on random variable $Y$, $g_X(x/y)$ can be computed as:

$g_X(x/y) = f(x, y)/f_Y(y)$

For every $y$, $g_X(x/y)$ is a probability function (pf/pdf) as a function of $x$.

Note: a conditional distribution is a mere collection of conditional probabilities. It is a valid probability distribution and all the rules of probability apply to it.

© 2018-19 Manjeet Dahiya