I Show that if X1 X2 X are independent random variables and

I. Show that if X1; X2; ? X are independent random variables and Y = c1X1 + c2X2 +... cpXp Then: sigma^2(Y) V(Y) = c1^2V(X1) + c2^2V(X2) +...cp^2V(Xp) where V(Z) is a variance of a random variable Z.

Solution

If X is a random variable then its variance is given by V(x)..

Similarly if X₁, X₂, ...., Xp are the independent random variable then their variances are given as V(X₁), V(X₂) ..... V(X_p)

By using variance property for constant functions i.e V(ax +b) = a² * V(x) where a,b are constants as Variance of constant alone is zero.

Y = c₁X₁ + c₂X₂ + .... +c_pX_p is a linear function of random variables where constants are in multiple of random variables

Therefore by using variance property V(Y) = c₁² V(X₁₎+c₂² V(X₂ ) + .... +c_p² V(X_p )