Tuesday, 26 July 2016

Conditional Probability Density Function

  Manoj       Tuesday, 26 July 2016

Let $X$ and $Y$ be two-dimensional random variables defined simultaneously on a sample space $S$. The joint probability function of $X$ and $Y$ is $ P(X=x, Y=y)$. Again the marginal probability function of $X$ and $Y$ are $g(x)$ and $h(y)$, where $g(x)>0$ and $h(y)>0$. Then the conditional probability function of $X$ given $Y=y$ is given by,
g(x/Y=y)=P(X=x,Y=y)h(y),h(y)>0.
Similarly, the conditional probability function of $Y$  given $X=x$ is given by
h(y/X=x)=P(X=x,Y=y)g(x),g(x)>0.
If $X$ and $Y$ are two dimensional continuous random variable, then
g(x/Y=y)=f(x,y)h(y),h(y)>0.
and h(y/X=x)=f(x,y)g(x),g(y)>0.
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Thanks for reading Conditional Probability Density Function

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