Indicator Variable

Description:

• Let random variable $I$ be indicator variable for event $A$, $I=1$ if $A$ occurs, 0 otherwise
• Then, for any random variable $Y$:
  • $E[X]=P(A)$
  • $E[X|Y=y]=P(A|Y=y)$
  • For discrete: $P(A)=\sum _{y}P(A|Y=y).P(Y=y)$
    • which is Conditional probability
  • For continuous: $P(A)={\int }_{-\infty }^{\infty }P(A|Y=y).f_Y(y)dy$
• When $X$ is an indicator variable for $n$ events:
  • $E[X^2]=E[X_1.X_2]=\sum _{i=1}^n\sum _{j=1}^{n}E[X_i.X_j]=\sum _{i=1}^{n}E[X_i^2]+\sum _{i\ne j}^{n}E[X_i.X_j]$ $=E[X]+2\sum _{i<j}^{n}E[X_i.X_j]$
    • $E[X_i^2]=E[X_i]$