doubts..please help ..

[radhika00]

X and Y are iid random variables. Suppose that X and Y are distributed normally with mean and variance.
Which of the following is true?
A. Random variable X- Y is identically zero.
B. Random variable X+ Y is equal to the random variable 2X.
C. Random variable X - Y has mean equal to 0 and std deviation equal to 2×sigma.
D. None of the above.

[Akshay Jain]

Correct option is D....
Suppose both X and Y have mean mu and variance sigma^2
E(X-Y) = E(X)-E(Y)=0 (expectation is a linear operator)
Var(X-Y)= E{(X-Y)-E(X-Y)}^2
=E(X-Y)^2= E(X)^2+E(Y)^2-2E(XY)
Now E(XY)=mu^2

[By using the covariance formula and the fact that X and Y are iid,i.e,  covariance is 0, E(XY)-E(X)E(Y)=0]
And sigma^2=E(X-mu)^2=E(X)^2 -mu^2

E(X)^2=sigma^2+ mu^2 and similarly for E(Y)^2
So E(X-Y)^2=2sigma^2 + 2mu^2 -2mu^2=2sigma^2
And std dev will b underoot of this, which is not eql to 2sigma