Conditional probability density function
Summary
Definition
- \(X,Y\) are two continuous random variables with joint probability density function \(f(x,y)\) and marginal probability density functions \(f_X(x)\) and \(f_Y\left( y \right)\) .
- The conditional probability density function of \(Y\) given \(X\) is defined as
\[f_{Y|X}\left( x \right)= \frac{f\left( x,y \right)}{f_X\left( x \right)}\]
- The conditional probability density function of \(X\) given \(Y\) is defined similarly as
\[f_{X|Y}\left( y \right)= \frac{f\left( x,y \right)}{f_Y\left( y \right)}\]
Example
- \(f\left( x,y \right)=x+y\) for \(0≤x≤ 1\) and \(0≤y≤ 1\) . Then \(f_X\left( x \right)=1/2 +x\) and
\[f_{Y|X}\left( x \right)= \frac{x+y}{1/2+x}\]
Conditional probabilities
\[P\left( X=x \right)=\int_{a}^{b}{ f_{Y|X}\left( x \right)dx }\]
Example (Continued)
\[P\left( X=x \right)=\int_{0}^{1/2}{ \frac{x+y}{1/2+x}dy }= \frac{1+4x}{4+8x}\]
\[P\left( X= \frac{1}{4} \right)= \frac{1}{3}\]