Bivariate Distribution
Table of contents
Joint Mass Function
Let $X$ and $Y$ be two discrete random variables.
- Discrete case: joint mass function
$$ p_{X,Y}(x,y) = P(X=x, Y=y) $$
Joint Densitity Function
Let $X$ and $Y$ be two continuous random variables.
- Continuous case: joint density function
$$ f_{X,Y}(x,y) $$
Remember, unlike mass functions, density functions are not probabilities.
Marginal Mass Function
If $X$ and $Y$ are joint distributions with joint function $p_{X,Y}(x,y)$,
- Discrete case: marginal mass function
$$ p_X(x) = P(X=x) = \sum_{y} p_{X,Y}(x,y) $$
Marginal Density Function
If $X$ and $Y$ are joint distributions with joint density function $f_{X,Y}(x,y)$,
- Continuous case: marginal density function
$$ f_X(x) = \int f_{X,Y}(x,y) dy $$
Independence
Let $X$ and $Y$ be two random variables.
$X$ and $Y$ are independent if:
$$ P(X=x, Y=y) = P(X=x)P(Y=y) $$
With Density Functions?
Even though density functions are not probabilities, if and only if the following holds for continuous random variables:
$$ f_{X,Y}(x,y) = f_X(x)f_Y(y) $$
then $X$ and $Y$ are independent.
Furthermore
Furthermore, if we can factorize a joint density function $f_{X,Y}(x,y)$ into,
\[f_{X,Y}(x,y) = g(x)h(y)\]where $g(x)$ and $h(y)$ are not necessarily density functions, then $X$ and $Y$ are independent.