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given E(X) , V(X) , E(Y) , V(Y) , COV(X,Y)

how to find the value of these two questions with the give data

a)E(X + 2Y) =?

b)VAR[X-2Y+1] =?

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Given E(X) , V(X) , E(Y) , V(Y) , COV(X,Y)

As we know :

Expectation follows linearity of expression (meaning each term can be separated)

So here :   E(X + 2Y)   =  E(X)  + E(2Y)

                                  =  E(X)  + 2 E(Y)  [ Follows from the simple fact : If we multiply each element of Y by 2 , then expectation(mean) is also going to be multiplied by 2 ].

Now    : for  Var(X - 2Y + 1)   , we need to know :

Variance only gets affected by scale factor e.g. in -2Y , scale factor is -2 and not by constant terms i.e. '1' here..In other words constant terms do not contribute to variance.

So general formula for Var(aX + bY + c)    =  a2 Var(X)  +  b2 Var(Y)  + 2ab Cov(X,Y) where Cov(X,Y) is covariance of X and Y[Var(X) is multiplied by a2 because standard deviation will scale by |a| and hence variance which is square of standard deviation will be squared as well ]

Hence   Var(X - 2Y + 1)      =       12 Var(X)  + (-2)2 Var(Y)  + 2(1)(-2) Cov(X,Y)

                                        =       Var(X)  + 4 Var(Y)  - 4 Cov(X,Y)   

Cov(X,Y)  is found   as   :   E(X.Y)   -   E(X).E(Y)  . Cov(X,X) is nothing but Var(X)..

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