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Let X and Y be two independent random variables. Suppose, we know that Var(√5X − √2Y) = 15 and Var(−√2X + Y) = 6.5. Var(X) and Var(Y) are:

(A) Var(X) = 2.5, Var(Y) = 2

(B) Var(X) = 2, Var(Y) = 2.5

(C) Var(X) = 3, Var(Y) = 1.5

(D) Var(X) = 1.5, Var(Y) = 2.5

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For this question , it should be kept in mind that if Var(X) and Var(Y) be the variances of X and Y random variables and a and b be constants , then :

Var ( aX + bY ) = a2 Var(X)  + b2 Var(Y) + 2ab Covar(X,Y)  where Covar(X,Y) is the covariance between the 2 variables used.

Also if X and Y are independent , then Covar(X , Y) = 0 , hence the above equation for the purpose of the given question is reduced to :

Var ( aX + bY ) = a2 Var(X)  + b2 Var(Y)

Now given ,

Var(√5X − √2Y) = 15     ⇒    5 Var(X) + 2 Var(Y)   =  15

and  Var(−√2X + Y) = 6.5.  ⇒ 2 Var(X) + Var(Y)    =   6.5

Hence solving these equations for Var(X) and Var(Y) , we get

Var(X)  =  2

Var(Y)  = 2.5

Hence B) is the correct option.