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Problem 1. Convergence in probability 8 points possible (graded) For each of the following sequences, determine...
8-31 Determine whether the series - converges or diverges. If it converges, find the sum. (If...
8-31 Determine whether the series - converges or diverges. If it converges, find the sum. (If the quantity diverges, enter DIVERGES.) Son 8-31 n=1 - = nsion Determine whether the series converges absolutely, conditionally, or not at all. (-1) - 1 n1/2 n=1 The series converges absolutely. The series converges conditionally. The series diverges. For which values of x does (n + 4)!x converge? n = 0 (-0,00) (-1,1) O no values exist O x = 0 (-4,4) Find the...
number 3 please Hw4.1708.pd 1 2 TL (2) LP convergence vs. convergence in probability Let Xn,...
number 3 please
Hw4.1708.pd 1 2 TL (2) LP convergence vs. convergence in probability Let Xn, nNbe a sequence of random variables and let X be another random variable. Given l < p < oo, we say that Xn converges to X in Lp if E(Xn-X") → 0 as n → x Show that this implies that Xn converges to X in probability (3) Monte Carlo Let f : 10, 1] → R be continuous and let Xn, n on...
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, respectivel...
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...
Problem 2. (The Convergence of Extreme Value) Let X1, X2, ... be i.i.d sample from the...
Problem 2. (The Convergence of Extreme Value) Let X1, X2, ... be i.i.d sample from the distribution with density function as: f(x) = >1 10 otherwise Define Mn = min(X1, X2, ... , Xn), answer the following questions. 1) Show that Mn P 1 as n +0. 2) Show that n(Mn – 1) converges in distribution as n + 00. Find out the limit distri- bution.
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e...
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
8. Let X be a continuous random variable with mgf given by It< 1 M(t)E(eX) 1...
8. Let X be a continuous random variable with mgf given by It< 1 M(t)E(eX) 1 - t2 (a) Determine the expected value of X and the variance of X [3] (b) Let X1, X2, ... be a sequence of iid random variables with the same distribution as X. Let Y X and consider what happens to Y, as n tends to oo. (i) Is it true that Y, converges in probability to 0? (Explain.) [2] (ii) Explain why Vn...
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives...
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? [10]
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives...
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X10 a. Compute the expected value E(X) and variance V(X) of X. [5] b. What is the probability function of X? (10)
2 8. For each i = 1, 2, ..., 10, Xi is a random variable that...
2 8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? (10)
Suppose X1, X2,... are independent Geometric (number of trials) random variables where Xi ~ Geome...
Suppose X1, X2,... are independent Geometric (number of trials)
random variables where Xi ~ Geometric(p = 1/i^2)
a) It is easily shown that Xn converges to a for some constant
a. Name it.
b) According to the Borel-Cantelli Lemmas, does Xn almost surely
converge to a?
Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
8-31 Determine whether the series - converges or diverges. If it converges, find the sum. (If the quantity diverges, enter DIVERGES.) Son 8-31 n=1 - = nsion Determine whether the series converges absolutely, conditionally, or not at all. (-1) - 1 n1/2 n=1 The series converges absolutely. The series converges conditionally. The series diverges. For which values of x does (n + 4)!x converge? n = 0 (-0,00) (-1,1) O no values exist O x = 0 (-4,4) Find the...
number 3 please
Hw4.1708.pd 1 2 TL (2) LP convergence vs. convergence in probability Let Xn, nNbe a sequence of random variables and let X be another random variable. Given l < p < oo, we say that Xn converges to X in Lp if E(Xn-X") → 0 as n → x Show that this implies that Xn converges to X in probability (3) Monte Carlo Let f : 10, 1] → R be continuous and let Xn, n on...
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...
Problem 2. (The Convergence of Extreme Value) Let X1, X2, ... be i.i.d sample from the distribution with density function as: f(x) = >1 10 otherwise Define Mn = min(X1, X2, ... , Xn), answer the following questions. 1) Show that Mn P 1 as n +0. 2) Show that n(Mn – 1) converges in distribution as n + 00. Find out the limit distri- bution.
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
8. Let X be a continuous random variable with mgf given by It< 1 M(t)E(eX) 1 - t2 (a) Determine the expected value of X and the variance of X [3] (b) Let X1, X2, ... be a sequence of iid random variables with the same distribution as X. Let Y X and consider what happens to Y, as n tends to oo. (i) Is it true that Y, converges in probability to 0? (Explain.) [2] (ii) Explain why Vn...
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? [10]
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X10 a. Compute the expected value E(X) and variance V(X) of X. [5] b. What is the probability function of X? (10)
2 8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? (10)
Suppose X1, X2,... are independent Geometric (number of trials)
random variables where Xi ~ Geometric(p = 1/i^2)
a) It is easily shown that Xn converges to a for some constant
a. Name it.
b) According to the Borel-Cantelli Lemmas, does Xn almost surely
converge to a?
Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
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