# Jointly Distributed Random Variables

Posted by Beetle B. on Wed 07 June 2017

## Probability

Given two random variables $$X,Y$$, the joint pdf is given by $$p(x,y)=P(X=x,Y=y)$$.

Let $$A$$ be an event. Then the joint pmf is:

\begin{equation*} P[(X,Y)\in A]=\sum\sum_{(x,y)\in A}p(x,y) \end{equation*}

The marginal pdf of $$X$$ is denoted by $$p_{X}(x)$$:

\begin{equation*} p_{X}(x)=\sum_{y}p(x,y) \end{equation*}

The joint density function is:

\begin{equation*} \int\int_{A}f(x,y)\ dxdy \end{equation*}

The marginal pdf is:

\begin{equation*} f_{X}(x)=\int_{-\infty}^{\infty}f(x,y)\ dy \end{equation*}

## Independence

Two random variables $$X$$ and $$Y$$ are independent if $$\forall(x,y),p(x,y)=p_{X}(x)p_{Y}(y)$$

I think the author makes the claim that to be independent, $$f(x,y)$$ must be of the form $$g(x)k(y)$$ and the region of positive density must be a rectangle aligned with the axes.

Multiple random variables are independent if they are independent for all subsets of $$X_{1},\dots,X_{n}$$.

## Conditional Probability

Let $$X,Y$$ be two random variables. The conditional pdf of $$Y$$ given $$X=x$$ is:

\begin{equation*} f_{Y|X}(y|x)=\frac{f(x,y)}{f_{X}(x)} \end{equation*}