Thursday, 26 April 2012

Independence and UnCorrelated

We Know that if Two Random variables $X$ and $Y$ are  Independent Statistically, Then

$$ E(XY)=E(X)E(Y)$$, i.e., They are UnCorrelated. It Doesnt Mean that Dependent Random Variables are Always Correlated. Lets Think of Some Examples with Dependent Random Variables being UnCorrelated.

1 comment:

  1. Let $X$ be any Random Variable with pdf having Even Symmetry i.e.,

    $$ f_X(-x)=f_X(x) $$, Then
    $$ E(X^{2k+1})=0, \forall k \in \mathbb Z_{\ge 0}$$
    Now if random variable $$Y=X^{2k} \forall k \in \mathbb Z_{\ge 0}$$,i.e., $Y$ is Dependent on $X$ , Then

    $$ E(XY)=0 $$ From above. Also $E(X)=0$ for $k=0$, so $$E(X)E(Y)=0$$

    so $$ E(XY)=E(X)E(Y)$$