Is it true that if two indicator variables are independent then they are uncorrelated?
If covariance =0 $\Rightarrow$ uncorrelated
Does the covariance between indicator variables exist?
thx
Is it true that if two indicator variables are independent then they are uncorrelated?
If covariance =0 $\Rightarrow$ uncorrelated
Does the covariance between indicator variables exist?
thx
It's true if there are only two of them; it's not true for more than two.
The proof is simple: Suppose $ X=\begin{cases} 1 & \text{with probability }p, \\ 0 & \text{with probability } 1-p, \end{cases} $ $ Y=\begin{cases} 1 & \text{with probability }r, \\ 0 & \text{with probability } 1-r. \end{cases} $ They are independent if and only if $\Pr(X=1\ \&\ Y=1)=pq$. The covariance is $ E(XY) - E(X)E(Y) = E(XY) - pq. $ Notice that $E(XY) = 0\cdot\Pr(XY=0) + 1\cdot\Pr(XY=1) = \Pr(XY=1)$, and that $XY=1$ if and only if $X=1$ and $Y=1$. So if the covariance is $0$ then $\Pr(X=1\ \&\ Y=1)=pq$.
The simplest instance of the fact that it doesn't work for more than two random variables is to let $Z$ be the mod-$2$ sum of $X$ and $Y$ and let $p=q=1/2$. Then $X$, $Y$, and $Z$ are pairwise independent, but clearly not independent. Hence uncorrelated but not independent.