Suppose that 1/2 where Z is any random variable with E22c, say, with c> 0 and...
Suppose that 1/2 where Z is any random variable with E22c, say, with c> 0 and a E R fixed, and X is any other random variable. (a) Let e > 0. Use Chebyshev's inequality to show that (b) For what values of does the argument in part (a) prove that Xn converges in probability to X? (c) For the values of α identified in part (b), what other mode of convergence of Xn to X is assured (without any...
Part c please Suppose that Zn n/2 where Zn is any random variable with Eena, say, with c0 and a E R fixed, and X is any other random variable. (a) Let > 0. Use Chebyshev's inequality to show that (b) For what values of o does the argument in part (a) prove that X converges in probability to X? (c) For the values of α identified in part (b), what other mode of convergence of X, to X is...
chebyshev’s inequality Problem 2 Chebyshev's Inequality Suppose that the random variable ? has a Poisson distribution with the parameter ? > 0, ~ ?(A). Using the Chebyshev's inequality prove that Problem 3 - Application of the Chebyshev's Inequality Suppose that a player plays a game where he gains a dollar with the probability or loses a dollar with the probability . That is, his gain from one game can be modeled as a random variable fi, such that If the...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
5) Let X be a random variable with mean E(X) = μ < oo and variance Var(X) = σ2メ0. For any c> 0, This is a famous result known as Chebyshev's inequality. Suppose that Y,%, x, ar: i.id, iandool wousblsxs writia expliiniacy" iacai 's(%) fh o() airl íinic vaikuitx: Var(X) = σ2メ0. With Υ = n Ση1 Y. show that for any c > 0 Tsisis the celebraed Weak Law of Large Numben
2. Suppose that is an exponential random variable with pdf f(y)= e), y>0. a. Use Chebyshev's Inequality to get an upper bound for the probability that takes on a value more than two standard deviations away from the mean. b. Use the given pdf to compute the exact probability that takes on a value more than two standard deviations away from the mean.
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...
(i) Show that 15 (ii) Show that (X) 5/12 and E(Y) 5/8 3(1 - 2X2 +X4) 4(2- 3X +X3) (iii) Show that 3(y|X) (iv) Verify thatE(Y)E(Y) 14] 7. (a) State Chebyshev's inequality and prove it using Markov's inequality 15] (b) Let (2, P) be a probability space representing a random experiment that can be repeated many times under the same conditions, and let A C S2 be a random event. Suppose the experiment is repeated n times (i) Write down...
Exercise 2 Consider a random variable X with E]5 and VarX 16 (a) Calculate P(lz-5 < 6) if X follows a normal distribution. (b) Use Chebyshev's inequality to provide a lower bound for P(-5). (No longer assume X is normal.)
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does there exist a random variable X:82 → R such that Xn-,X as n →oo? Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does...