Joint probability mass function

Definition:

• Jointly distributed random variable for Probability mass function
• When $X$ and $Y$ are Discrete random variables, define the joint probability mass function of $X$ and $Y$ by $p(x,y)=P(X=x,Y=y)$
  • $p_X(x)=\sum _{j}p(x,j)$
  • $p_Y(x)=\sum _{i}p(x_i,y)$

Independent random variables

• If the random variable $X$ and $Y$ for any two sets of real numbers $A$ and $B$ $P\{X\in A,Y\in B\}=P\{X\in A\}.P\{Y\in B\}$
  • For discrete, $p(x,y)=p_X(x).p_Y(y)\text{ for all }x,y$
  • $P\{X\le A,Y\le B\}=P\{X\le a\}.P\{Y\le b\}$
  • $F(a,b)=F_X(a).F_Y(b)\text{for all }a,b$
• Two discrete random variables are independent if and only if their JPMF can be expressed as: $f_{X,Y}=h(x).g(y)-\infty <x,y<\infty$

Expectation of function of joint discrete variable:

• If $X$ and $Y$ have a Joint probability mass function $p(x,y)$, then $E[g(X,Y)]=\sum _y\sum _{x}g(x,y)f(x,y)$