The joint probabillity mass function

Summary

• If \(X_1,…,X_n\) are \(n\) discrete random variables then the joint probabillity mass function ( joint pmf) \(f: R^n→[0,1]\) is defined as

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

• The joint cumulative distribution function \(F:R^n→[0,1]\) is defined as

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

Example

• \(S=\{ α,β,γ, θ \}\) (equally likely)
• \(X_1\) is defined by \(X_1\left( α \right)=2\) , \(X_1\left( β \right)=2\) , \(X_1\left( γ \right)=2\) and \(X_1\left( θ \right)=0\) .
• \(X_2\) is defined by \(X_2\left( α \right)=0\) , \(X_2\left( β \right)=1\) , \(X_2\left( γ \right)=1\) and \(X_2\left( θ \right)=1\) .
• Then \(f\left( 2,1 \right)=1/2\) , \(f\left( 0,0 \right)=0, F\left( 1,1 \right)=1/4\) .