Suppose that a sequence of mutually independent and identically distributed discrete randomvariables X₁, X₂, X3, X has the following probability density functionf(x: 0) =a)b)c)6e-0x!for x = 0,1,2,...0,elsewhereShow that for any e > 0 and S, X₁, lim P(|S, -01 ≥ £) = 0.Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8.Let 8₁ = X₁ +2x+2x₁-X and 6₂ = (X₁ + X₂ + X₁ + X₁) be two unbiased estimators of 8. Whichone of the two estimators is more efficient?d) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of theparameter @?e) Use the one-parameter regular exponential family definition to find the functions,

As per bartleby guidelines we can solve only first three subparts and rest can be reposted

Answered: a) b) c) Show that for any >0 and S,…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose that X is a continuous random variable with density function f(x). If f(x)=k for −5≤x≤3 and f(x)=0 otherwise, determine the value of k.
Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?
Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.
Suppose that two-dimensional continuous random variable (X, Y) has joint probability density function given by f(x,y) = 24xy, x is less than equal to 1 and greater than equal to 0, y is less than equal to 1 and greater than equal to 0, x+y is less than equal to 1 and greater than equal to 0. Check that E(Y) = E[E(Y|X)] and V(Y) = E[V(Y|X)] + V[E(Y|X)].
Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
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Suppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]
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Suppose that ƒ is a uniform joint probability density function on0 ≤ x 6 2, 0 ≤ y < 3. What is the formula for ƒ? What is theprobability that X < Y?
suppose that the probability density function of x is f(x)={3x^2, 0, 0<x<1 elsewhere. determine p(x<1/3), P(1/3 <=x<2/3), and P(x=>2/3)
Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).
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a) b) c) Show that for any >0 and S, X₁. lim P(|S, -612 ) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁=X₁2X+2X-X₁ and 6₂ = (X₁ + X₂ + X₁ + X₂) be two unbiased estimators of 6. W one of the two estimators is more efficient?