Answer :

Correct answer is : -1

Given: E(X) = E(Y) = 0.5

$\frac{1}{{\lambda }_1}=\frac{1}{{\lambda }_2}=0.5$

λ₁ = λ₂ = 2

$Var\left(X\right)=\frac{1}{{\lambda }_1^2}=\frac{1}{2^2}=0.25$

$Var\left(Y\right)=\frac{1}{{\lambda }_2^2}=\frac{1}{2^2}=0.25$

Also it is given, Z = X + Y

And Var (Z) = 0

∴ Var (X + Y) = 0

Var X + Var Y + 2 Cov (X, Y) = 0

On putting values :

0.25 + 0.25 + 2 Cov (X, Y) = 0

$Cov\left(X,Y\right)=-\frac{0.5}{2}=-0.25$

Correction coefficient is given by :

$r=\frac{Cov\left(X,Y\right)}{\sqrt{Var\left(X\right)Var\left(Y\right)}}=\frac{-0.25}{\sqrt{0.25\times 0.25}}=-1$