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Let X and Y be continuous random variables with joint distribution function F(x, y), and let...

Let X and Y be continuous random variables with joint distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be functions of X and Y . Prove the following:

(a) E[cg(X, Y )] = cE[g(X, Y )].

(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].

(c) V ar(a + X) = V ar(X).

(d) V ar(aX) = a 2V ar(X).

(e) V ar(aX + bY ) = a 2V ar(X) + b 2V ar(Y ) + 2abCov(X, Y ).

(f) If X and Y are independent, then E[XY ] = E[X]E[Y ].

(g) If X = Y , then Cov(X, Y ) = V ar(Y ).

(h) Cov(X, Y ) = E[XY ] − E[X]E[Y ].

(i) Independence of X, Y =⇒ Cov(X, Y ) = 0.

(j) Cov(X, Y ) = 0 =6⇒ Independence of X, Y .

Math   Statistics-And-Probability   
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