How does one calculate the expected value? Suppose a fair coin is flipped twice. Define two random variables $X$ and $Y$ , where $X$ counts the number of heads, and $Y$ counts the number of tails, in the two flips.


*

*Evaluate $\Bbb E(X), \Bbb E(Y)$. 

*From these values, can you decide whether $X$ and $Y$ are dependent or independent?


These expected values will be the same if the coin is fair and the two flips are independent. For example X=2 for HH only, X=1 for HT and TH, and X=0 for TT only. That gives 2*(1/4)+1*(1/2)+0*(1/4)=1 for E(X). If you do it, E(Y) also comes out 1. 
But from these alone one cannot conclude whether X,Y are independent, and they are not, since given X=2 it must be that Y=0.
