- X is a Gaussian random variable N (m, o). Find the density of X=a. X+h

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Exercise 40 Let X Unif(0,1). Let g(x) = e" and Y = g(X). (i) Find the density fx of the random variable X. (ii) Find the cdf Fy (y) = P(Y < y) of Y. (iii) Find the density fy of the random variable Y.
Find the expected value of the function g(X) = X' where X is a random variable defined by the density fx(x) = (1/2)u(x)e(-x/3). Select one: O a. 243 O b. none of these O c. 48 O d. 162 REDMI NOTE 9S AI QUAD CAMERA
Consider a random variable Y with density S (1/2)(y + 1), -1 < y < 1, else. f(y) Calculate E(W) and V(W) for the random variable W = 4+Y – 2Y².
Q 4.2. Let (X, Y) be a random variable with the following density: {152-2 0 £x,x (x, y) = { 1. Express E(Y|X) in terms of X. 2. Express Var(XY) in terms of Y. 0 < x < y < 1, otherwise.
(d) Let X, Y be independent Poisson random variables with X~ Poisson(1.5), Y~ Poisson (0.5). Compute the probability P(X + Y ≤ 2). Express your answer up to 3rd decimal place. (You shall use calculator to complete this problem.) Suppose two random variables X, Y admits density f(x, y) = 9 -x-3y 7(2x+y)e¯ " x, y ≥ 0, and f(x, y) = 0 for other (x, y). Compute the covariance of X, Y. Use the result to decide which of the following holds: Var[X + Y] .
3. Let random variables X and Y be independent with joint density fe(x, y). Let Ix (0) and Iy (0) be the Fisher information of X and Y, respectively. Prove that the Fisher information I(x,y)(0) = Ix(0) + Iy(0).
b) Let Y1, Y2.,Y5 be independent random variables with probability density function y 1 e 4 4 f(y)= ,y > 0 ,elsewhere 3 Determine the distribution and parameter of V = EY; using the method of moment i=1 generating function. Hence, find the mean of V using the moment generating function.
Find the density of U = Y1 +Y2, where Y1 and Y2 are independent random variables with densities (yı – 1), 3 < Yı < 4, (y2 + 2), 0 < Y2 < 1, fy, (y1) : fy,(y2) = otherwise, otherwise.
c) Let Y₁, Y₂,..., Yn be a random sample whose probability density function is given by f(v:B)= 684 - fa 00 0, elsewhere 200 200 200 and suppose that n = 200, y = 20, y = 100, y = 250 and $ = 0.025. i=1 i) Derive the standard error of ß, se(B) = 0.0009, using MLE approach. ii) Find an approximate 95% Confidence interval for B.
Let Y₁, Y₂,..., Yn be a random sample from the inverse Gaussian distribution with probability density function: f(y, μ, 2) = { Where μ and are unknown. a) What are i. 1 -ACT √ λ 2ny3)že elsewhere if y > 0 the likelihood ii. the log likelihood functions of u and λ. b) Find the maximum likelihood estimators of u and 2.

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X is a Gaussian random variable N (m, o). Find the density of Y=a X+h