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Joint Distribution

When a pair $ (X,Y)$ of random variables is considered, a joint density function $ f(x,y)$ is used to compute probabilities constructed from the random variables $ X$ and $ Y$ simultaneously. Given the joint density function $ f(x,y)$, the distribution for each of $ X$ and $ Y$ is called the marginal distribution. The marginal density functions of $ X$ and $ Y$, denoted by $ f_X(x)$ and $ f_Y(y)$, are given respectively by

$\displaystyle f_X(x) = \int_{-\infty}^\infty f(x,y) dy$    and $\displaystyle \quad
f_Y(y) = \int_{-\infty}^\infty f(x,y) dx.

Conditional probability distributions. Suppose that two random variables $ X$ and $ Y$ has a joint density function $ f(x,y)$. If $ f_X(x) > 0$, then we can define the conditional density function $ f_{Y\vert X}(y\vert x)$ given $ X = x$ by

$\displaystyle f_{Y\vert X}(y\vert x)
= \frac{f(x,y)}{f_X(x)}.

Similarly we can define the conditional density function $ f_{X\vert Y}(x\vert y)$ given $ Y = y$ by

$\displaystyle f_{X\vert Y}(x\vert y)
= \frac{f(x,y)}{f_Y(y)}

if $ f_Y(y) > 0$. Then, clearly we have the following relation

$\displaystyle f(x,y) = f_{Y\vert X}(y\vert x) f_X(x) = f_{X\vert Y}(x\vert y) f_Y(y).

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