The following are general questions about covariance and correlation (each is a
ID: 3151082 • Letter: T
Question
The following are general questions about covariance and correlation (each is a seperate question).
(a) Let X,Y be random variables. Show that Var[X+Y] = Var[X] + Var[Y] + 2Cov(X,Y).
(b) Suppose that X = X1+ ... + Xn and Y = Y1+ ... + Yn (Xi, Yi r.v.'s). Show that:
(d) Let X be a random variable. Is it possible that pX, X2 = 1 for some X? What about pX, X2 = -1?
(e) Let X,Y be random variables with the following joint distribution (with R > 0):
Show that X,Y are uncorrelated, but that they are not independent.
Explanation / Answer
1) each part is a different question and need to be solved seperately, i m giving u the solution of 1st.
in this question we need to prove, Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y )
as we know that
Var(X + Y ) = E[(X + Y )(X + Y )] E[X + Y ]2
= E[X^2 + 2XY + Y^2] (µx + µy)^2
= E[X^2 + 2XY + Y^2] µ^2x 2µxµy µ^2y
= (E[X^2] µ^2x) + (E[Y^ 2] µ^2y) + 2(E[XY ] µxµy)
= Var(X) + Var(Y ) + 2Cov(X, Y )
hence we get the desired result
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