Question

2. Explain in words, and words only, the following properties of expected values. NOTE: X and Y are random variables and k is a constant. (a) E(k) = k (b) E(X+Y) = E(X) + E(Y) (c) E(X/Y) + E(X)/E(Y) (d) E(X+Y) E(X)*E(Y) (unless what?) (e) E(X2) # (E(X)]? (1) E(kX) = E(X) 3. For random variable X with mean H. variance is defined var(X) = Ef(X-M.)'. Show how variance can be expressed only in terms of E(X) and E(X). 4. Explain in words, and words only, the following properties of variance. NOTE: X and Y are random variables and k is a constant. (a) var(k)=0 (b) var(X+Y) = var(X) + var(Y) (c) var(X-Y) = var(X) + var(Y) (d) var(kx) k var(X) 5. For random variable X with mean and random variable y with mean y covariance is defined cov(X) = E(X - Y - My)]. Show how covariance can be expressed only in terms of E(XY) and the means.

Answer 1

Solution 2 X and y be random variables @ Go= Elk)= K ElkJ = Ekip (x=k) : = K [ plaz) is equal to 1 Sum of prob - K1 je [plx=k)=1 Elkj=K) ElX+Y) = E(X+E 11) O E(X[4) + ElXJ it is not true.). Elx*y J + Elx) · Ely because X and Y are not independent rondom variables. © E(x2)+ ( E(X)]? JE(X)=42. Elxal itis - i mean I hade ElkW = K El X) Where E(X2) és second Elkx) = {ka. PlX=K) a sol moment Ek E 7.plx=k) ?'Elx) = {X.P(x=k] ( E(ky = K.EIX) 7