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If two random variables X and Y are such that $$E(X)+E(Y)=0\dots, \tag1$$ $Var(X)=Var(Y)$ and $1+r=0$ where $r$ is the correlation co-efficient between $X$ and $Y$,then what is the relation between $X$ and $Y$?[E(x) is the expectation of X]

I do not really know how to proceed. This is all that I could gather:

$Var(X)=Var(Y)\implies E(X^2)-E^2(X)=E(Y^2)-E^2(Y)$ which means that $E(X^2)=E(Y^2)$ on account of $(1)$.

From the other condition I get $r=-1$ ie.e $E(XY)-E(X)E(Y)=-Var(X)=-Var(Y)$.

But after that I seem lost;can anyone please point out how I can establish a relation between $X$ and $Y$?

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The correlation can be writen as: $$\frac {E(XY)-E(X)E(Y)}{\sqrt {Var(x)}\sqrt {Var(y)}}=-1$$ So $E(X^2)=E(Y^2)$ give us that $$\frac {E(XY)+E^2(X)}{Var(X)}=-1$$. Does this help? –  HipsterMathematician Nov 10 '12 at 15:01

1 Answer 1

up vote 2 down vote accepted

$\begin{align} Var[Y+X]&=Var[Y]+Var[X]+2Cov[X,Y]\\ &=Var[Y]+Var[X]-2\sqrt{Var[X]Var[Y]} \; \because \text{X,Y are negatively correlated} \\ &=0 \; \because \text{Var[X]=Var[Y]} \end{align} $

A Random variable with zero variance is a constant.

$\therefore X+Y=c$

but $\because E[X]+X[Y]=0$ we have that $ E[X+Y]=E[X]+E[Y]=E[c]=c=0$.

$\therefore X=-Y$

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I didn't understand why $X[Y]$ ? –  HipsterMathematician Nov 10 '12 at 16:00
    
@Charlie sorry typo –  Amatya Nov 10 '12 at 16:07
    
Ah! Okay, thanks! –  HipsterMathematician Nov 10 '12 at 16:07

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