2. Suppose one has n non-degenerate random variables X1, X2,, Xn so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.
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- A coin is tossed twice. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find (a) P( W=2 | Z= 1) (b) P( Z=0 | W = 1) ; (c) are Z and W independent?Let A, B, and C be independent random variables, uniformly distributed over [0,3], [0,2], and [0,4] respectively. What is the probability that both roots of the equation Ax2Bx+C = 0 are real?Let X be a random variable defined by 2 3 P(X = x) k 2k 2k 2k 2k Then E(X) and V(X) are equal to: None of these O E(X)=13/5 and V(X)=41/25 O E(X)=20/9 and V(X)=140/81 O E(X)=27/11 and V(X)=206/121
- show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.Let X1, X2,..., X3 denote a random sample from a population having mean u and variance o?... Which of the estimators have a variance of 7 X1+X2++X, 7 2X1-X6+X4 2 3X1-X3+X4 2 2(X1+X2+.+X¬) 4 7
- 9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.3. If X;~N(0,1), i = 1,2,3,4 are independent Normal random variables and Y is given as Y =(X,+X,)' +(X, + X,)' . If aY~x², . find the constant a. 22. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
- B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.Suppose X1,X2.X10 are independent N(H,02) random variables. Z1,Z2.Z5 are independent N(0,1) random variables. The distribution of X + Z is a. N(u, O b. None of these С. N(u,- 10 d. N(0,4 Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli random variable, independent of X, that returns 1 if the tulip survives longer than 30 days, and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that returns the remainder of the division of X +Y by 2. For example, if X = 1 and Y = 0, Z = remainder() = 1 (a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2. (b) Prove that X, Y, Z are pairwise independent but not mutually independent. (c) By computing Var[X+Y+Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that Var[X+Y+Z] = Var[X]+Var[Y] +Var[Z] %3D (observe that this also follows from the proposition on slide 5 of the lecture segment entitled "Binomial distribution").