tag:blogger.com,1999:blog-5967982571807204356.post2264126362910981822..comments2015-12-17T20:53:47.229-08:00Comments on Mathematics in Research: Independence and UnCorrelatedekaveerahttp://www.blogger.com/profile/12705417398875109678noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-5967982571807204356.post-40511336380067163162012-04-26T05:48:32.956-07:002012-04-26T05:48:32.956-07:00Let $X$ be any Random Variable with pdf having Eve...Let $X$ be any Random Variable with pdf having Even Symmetry i.e.,<br /><br />$$ f_X(-x)=f_X(x) $$, Then<br />$$ E(X^{2k+1})=0, \forall k \in \mathbb Z_{\ge 0}$$<br />Now if random variable $$Y=X^{2k} \forall k \in \mathbb Z_{\ge 0}$$,i.e., $Y$ is Dependent on $X$ , Then<br /><br />$$ E(XY)=0 $$ From above. Also $E(X)=0$ for $k=0$, so $$E(X)E(Y)=0$$<br /><br />so $$ E(XY)=E(X)E(Y)$$ekaveerahttps://www.blogger.com/profile/12705417398875109678noreply@blogger.com