9. Let X and Y be independent and identically distributed random variables with mean u and...
Let Yı, Y2, Ys, and Y4 be independent, identically distributed random variables from a mean u and a variance 02. Consider a different estimator of u: W=Y+Y2+2Y3+ Y 00 This is an example of a weighted average of the Y a) Show that W is a linear estimator. b) Is W an unbiased estimator of u? Show that it is - or it isn't (E(W) = Find the variance of W and compare it to the variance of the sample...
Let X and Y be independent identically distributed random variables with means µx and µy respectively. Prove the following. a. E [aX + bY] = aµx + bµy for any constants a and b. b. Var[X2] = E[X2] − E[X]2 c. Var [aX] = a2Var [X] for any constant a. d. Assume for this part only that X and Y are not independent. Then Var [X + Y] = Var[X] + Var[Y] + 2(E [XY] − E [X] E[Y]). e....
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
Let Y, Y2, Yz and Y4 be independent, identically distributed random variables from a population with mean u and variance o. Let Y = -(Y, + Y2 + Y3 +Y4) denote the average of these four random variables. i. What are the expected value and variance of 7 in terms of u and o? ii. Now consider a different estimator of u: W = y + y + y +Y4 This an example of weighted average of the Y. Show...
(1 point) If X and Y are independent and identically distributed uniform random variables on (0, 1), compute each of the following joint densities. (a) U -3X, V - 3X/Y. fu.v(u, v) - (b) U - 5X + Y, V - 3X/(X + Y)
Exercise 7. Let Xi, X2, . . . be independent, identically distributed rundorn variables uithEX and Var(X) 9, and let Yǐ = Xi/2. We also define Tn and An to be the sum and the sample mean, respectively, of the random variablesy, ,Y,- 1) Evaluate the mean and variance of Yn, T,, and A (2) Does Yn converge in probability? If so, to what value? 3) Does Tn converge in probability? If so, to what value? (4) Does An converge...
2. Let X1, X2,. . , Xn denote independent and identically distributed random variables with variance σ2, which of the following is sufficient to conclude that the estimator T f(Xi, , Xn) of a parameter 6 is consistent (fully justify your answer): (a) Var(T) (b) E(T) (n-1) and Var(T) (c) E(T) 6. (d) E(T) θ and Var(T)-g2. 72 121
Exercise 2 (2). Let X and ε be independent normally distributed random variables such that X∼N(5,4)andε∼N(0,9).LetY bearandomvariablegivenbyY =1+2X+ε.Compute: (a) E(Y ) (b) Var(Y ) (c) Cov(X, Y ) (d) Corr(X, Y ) (e) What is the value of the ratio Cov(X, Y )/Var(X) ? (f) If Y = 1 + 3X + ε instead, what would be the value of Cov(X, Y )/Var(X) ? (g) If Y = 1 + 7X + ε instead, what would be the value of...
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when 6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...