The joint probability density function

Summary

• \(X,Y\) are continuous random variables i f there is a joint probability density function (joint pdf) \(f:R^2→R\) such that

\[P\left( \left( X,Y \right)∈A \right)=\int_{A}{ f\left( x,y \right)dxdy }\]

• for all \(A∈R^2\) .
• The joint cumulative distribution function \(F:R^n→[0,1]\) is defined as

\[F(x_1,…,x_n)=P(X_1≤x_1,…, X_n≤x_n)\]

• If \(F\) is differentiable then the relationship between the joint pdf and the joint cdf is as follows:

\[f\left( x,y \right)= \frac{∂^2F\left( x \right)}{∂x∂y}\]

\[F\left( x,y \right)=\int_{-∞}^{x}{ \int_{-∞}^{y}{ f\left( u,v \right)dudv } }\]