Determine if the given statements in (1)-(10) are true or false (circle your answer).

1. For independent random variables X and Y , we have that Var(X+ Y ) = Var(X) + Var(Y ). 2. Let Y = 3X1 + 2X2 3X3 where X1,X2,X3 are independent and identically distributed with the normal distribution N(1,1). Then Y has the normal distribution with mean 2 and variance 4. 3. If n independent and identically distributed random variables Xi's have the Pareto distribu- tion with PDF f(x) = x2, x > 1, i = 1,...,n, then the minimum X1:n = min{X1,...,Xn} also has the Pareto distribution with PDF g(x) = nxn1, x > 1. 4. Let X1 and X2 be two non-negative, continuous random variable with CDF F(x) and G(x), respectively, and finite means E(X1)

9. If two random variabes X and Y are independent, then the conditional expectation satisfies that E(g(X)|Y ) = g(X). 10. Suppose that Yn Gn(y), n = 1,2,.... If Yn P Y as n where a random variable Y G(y), then limn Gn(y) = G(y), for all values y at which G(y) is continuous.

1. For independent random variables X and Y, we have that Var(X + Y) = Var(X) + Var(Y). True False 2. Let Y = 3X, + 2X2 -3X3 where X1, X2, X3 are independent and identically distributed with the normal distribution N(1, 1). Then Y has the normal distribution with mean 2 and variance 4. True False 3. If n independent and identically distributed random variables X/'s have the Pareto distribu- tion with PDF f(x) = x 2, x > 1, i =1,...,n, then the minimum X1: = min{ X1, . .., X, } also has the Pareto distribution with PDF g(x) = no "-1, x > 1. True False 4. Let X, and X2 be two non-negative, continuous random variable with CDF F(x) and G(x), respectively, and finite means E(X1) E(X 2). True False 5. Let Mx (t) be the MGF of independent and identically distributed random variables X1, . .., X, with finite mean / = E(X;) and variance o' = Var(Xi), i = 1,...,n. Consider now the standardization Z = An #, where X,, is the sample average. Then the MGF of Z, Mz(t) = e vn/o [Mx(no)]". True False 6. Let (X, Y) be a vector of two continuous random variables. Suppose that the conditional expectation E(Y[X = >) = x, and that X ~ EXP(1). Then we must have E(Y) = Ex [E(Y X)] = Ex(X] = 1. True False7. Let (X, Y) be a vector of two continuous random variables. Suppose that the conditional expectation E(Y X = x) = r, and that X ~ EXP(1). Then we must have Var(Y ) = Varx [E(Y|X)] = Varx [X] = 1. True False 8. Suppose that random variable X; has the MGF Mx. (t), i = 1, ...,n, respectively. Then the MGF of the maximum, Mmax,sign (x;) (t) = maxisisn Mx. (t). True False9. If two random variabes X and Y are independent, then the conditional expectation satisfies that E(g(X ) |Y) = 9(X). True False 10. Suppose that Y, ~ G.(y), n = 1,2, .... If Y, => Y as n - co where a random variable Y ~ G(y), then lim Gn(y) = G(y), 71 700 for all values y at which G(y) is continuous. True False