University of California, Los AngelesDepartment of StatisticsStatistics 100A Instructor: Nicolas ChristouHomework 7EXERCISE 1Show that V ar(X − Y ) = V ar(X) + V ar(Y ) − 2Cov(X, Y ).EXERCISE 2If X and Y are independent variables with equal variances find Cov(X + Y, X − Y ).EXERCISE 3If U = a + bX and V = c + dY . show that |ρU V| = |ρXY|.EXERCISE 4Let U and V be independent random variables with means µ and variances σ2. Let Z = αU + V√1 − α2.Find E(Z) and ρU Z.EXERCISE 5Suppose that X and Y are two independent measurements. Also it is given that E(X) = E(Y ) = µ, but σXand σYare unequal. The two measurements are combined by means of a weighted average to giveZ = αX + (1 − α)Ywhere α is a scalar and 0 ≤ α ≤ 1.a. Show that E(Z) = µ.b. Find α in terms of σXand σYto minimize V ar(Z).c. Under what circumstances is it better to use the averageX+Y2than either X or Y alone to estimateµ. Note: X, Y,X+Y2all have the same expected value (µ). Therefore we would preferX+Y2if itsvariance is smaller than the variance of X and the variance of Y .EXERCISE 6Suppose that Xi, where i = 1, ···, n, are independent random variables with E(Xi) = µ and V ar(Xi) = σ2.Let¯X =1nPni=1Xi. Show that E(¯X) = µ and V ar(¯X) =σ2n.EXERCISE 7Let T =Pnk=1kXk, where Xkare independent random variables with mean µ and variance σ2. Find E(T )and V ar(T ).EXERCISE 8Let X and Y have the following joint probability density function:fXY(x, y) =67(x + y)2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1a. Find the covariance and correlation of X and Y .b. Find E(Y |X = x) for 0 ≤ x ≤ 1.EXERCISE 9Three stocks A, B, C have the following expected (mean) returns and standard deviations:µ σA 20% 8%B 10% 4%C 15% 6%Also, the correlation coefficients are: ρAB= 0.5, ρAC= 0.2, and ρBC= −1. You have $100000 to invest.a. If you invest $75000 in stock A and $25000 in stock B what will be the expected (mean) return ofthis portfolio?b. What is the risk (variance) of the portfolio of part (a)?c. You are risk averse. Which portfolio has less risk:i. The portfolio of part (a)?ii. A portfolio in which you invest $50000 in stock B, and $50000 in stock C?d. You equally allocate your funds into the three stocks A, B, C. What is the risk of this portfolio?Exercise 10Answer the following questions:a. If Z is a standard normal variable, what is Cov(Z, Z2)?b. If Z is a standard normal variable and if Y is defined by Y = a + bZ + cZ2, where a, b, and c areconstants show that ρ(Y, Z)