1) Let X₁, X2, X3, ... Xn be a sequence of independent and identically distributed with thefollowing probability density functionf(x; 0) = {(0 + ¹S(0+1)x-0-2if 1 < x < 00otherwiseb) Use method of moments to find a statistic that we can use to estimate theparameter 8.c) Check if Y = is an unbiased estimator of 0 or not, where ☎ is the estimatorobtained in b).d)Find the maximum likelihood estimator of the parameter 0.e) What is the fisher information in the sample of size n about the parameter 0.

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Answered: 1) Let X₁, X2, X3, ... Xn be a sequence…

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, 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 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 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?
If two random variables X1 and X2 have the joint density function given by f (x1, x2) = x1x2, 0 < x1 < 1, 0 < x2 < 2 0, otherwise Find the probability that (a) Both random variables will take on values less than 1 (b) The sum of the values taken on by the two random variables will be less than 1.
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 X and Y are independent and identically distributed (i.i.d.) randomvariables, each with the uniform distribution on [0, 1]. What is the cumulative distributionfunction and the density function of XY ?
2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.
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),
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).
6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?
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 that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independent

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