For any random variables X1 X2 and any numbers c1 c2 show th

For any random variables X1, X2 and any numbers c1, c2, show that

Solution

For any random variables X1 , X2 and any numbers c1 , c2 show that,

Var ( c1X1 + c2X2 ) = c1²Var(X1) + c2²Var(X2) + 2c1c2Cov(X1,X2)

This we want to prove,

Let us consider , V = c1X1 + c2X₂ ,

Put this values in equation Var (V) = E (V - E(V))² (by definition of variance)

Var(V) = E [ (c1X1 + c2X2 ) - E(c1X1 + c2X2 ) ]²

= E [ (c1X1 - E(c1X1) ) + (c2X2 - E(c2X2) ]²

  Now we will use the formula as,

   (a + b)² = a² + b² +2ab

Var(V) = E [ (c1X1 - E(c1X1) )² + (c2X2 - E(c2X2) )² + 2 ( (c1X1 - E(c1X1)) (c2X2 - E(c2X2)) )

taking expectation inside,

=    E (c1X1 - E(c1X1) )²  + E (c2X2 - E(c2X2)² + 2 E [ (c1X1 - E(c1X1) ) ( c2X2 - E(c2X2) ) ]

And we know that the result E (CX) = C E(X) apply this result to the equation

= c1² E(X1 - E(X1))² + c2² E(X2 - E(X2))² + 2c1c2 { E [ (X1 - E(X1) (X2 - E(X2) ] } ____a)

We know that,

E(X1 - E(X1))² = Var(X1)

and E(X1 - E(X1))² = Var(X2)

E [ (X1 - E(X1) (X2 - E(X2) ] = Cov(X1,X2)

Now put all these values in equation a) ==>

Var ( c₁X₁ + c₂X₂ ) = c1² Var(X1) + c2² Var(X2)