The joint probabillity mass function

Summary

  • If X1,…,XnX1,…,Xn are nn discrete random variables then the joint probabillity mass function ( joint pmf) f:Rn→[0,1]f:Rn→[0,1] is defined as

f(x1,…,xn)=P(X1=x1,…,Xn=xn)f(x1,…,xn)=P(X1=x1,…,Xn=xn)

  • The joint cumulative distribution function F:Rn→[0,1]F:Rn→[0,1] is defined as

F(x1,…,xn)=P(X1≤x1,…,Xn≤xn)F(x1,…,xn)=P(X1≤x1,…,Xn≤xn)

Example

  • S={α,β,γ,θ}S={α,β,γ,θ} (equally likely)
  • X1X1 is defined by X1(α)=2X1(α)=2 , X1(β)=2X1(β)=2 , X1(γ)=2X1(γ)=2 and X1(θ)=0X1(θ)=0 .
  • X2X2 is defined by X2(α)=0X2(α)=0 , X2(β)=1X2(β)=1 , X2(γ)=1X2(γ)=1 and X2(θ)=1X2(θ)=1 .
  • Then f(2,1)=1/2f(2,1)=1/2 , f(0,0)=0,F(1,1)=1/4f(0,0)=0,F(1,1)=1/4 .