Squeezing. Let S and X have density functions satisfying b(x) ≤ fs(x) ≤ a(x) and fs(x) ≤fx(x). Let U be uniformly distributed on [0, 1] and independent of X. Given the value X, weimplement the following algorithm:if Ufx (X) > a(X),otherwise: if Ufx (X)

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Answered: Squeezing. Let S and X have density…

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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Theorem A at Section 8.8.1 saysA necessary and sufficient condition for T(X1,..., Xn) to be sufficient for a parameter θ is that the joint probability function (density function or frequency function) factors in the formf(x1,...,xn|θ) = g[T(x1,...,xn),θ]h(x1,...,xn)
(2) Let X1, X2, X3, X4 have the joint probability density function f(x1, x2, x3, x4) = ( 24e −(x1+x2+x3+x4) , 0 < x1, x2, x3, x4 < ∞ 0, elsewhere Let Y1 = X1, Y2 = X2 − X1, Y3 = X3 − X2, Y4 = X4 − X3. (i) Using the change of variable technique, find the joint probability density function of Y1, Y2, Y3, Y4 (ii) Find the conditional distribution of Y4 given Y1, Y2, Y3.
Determine k so thatf(x, y) = kx(x − y) for 0 < x < 1, −x < y < x0 elsewherecan serve as a joint probability density.
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)
Let X, Y, and Z have the joint probability density function F(x,y,z) = { kx2 yz, 0 < x, y < 1, 0 < z < 2, a) Find k b. Find P(X < ¼, Y > ½, 1 < Z < 2)
Let X have pdff(x) = e-X /(1+ e -X)2 −∞< x < ∞Show that Y = 1/(1+ e -X)has the uniform distribution U(0,1). State the method you use and show allwork.
5. Suppose that the joint probability density function of X and Y is given by f(x, y) = C (x² + y), 0 < x < 2, 0 < y < 2 and f(x, y) = 0 otherwise. (a) Find C. (b) Find the marginal density functions of X and Y. (c) Compute P(X > Y). (d) Compute E[X]. (e) Compute E[Y].
: say in each case, if the function f is a probability density over interval I. a) f (x) = 0.5x + 0.5 on I = [-1; 1] b) f (x) = x on I = [0; 1] c) f (x) = 0.5 on I = [-1; 1] d) f (x) = -2x² + 2 over I [0; 1]
Suppose that g is an easy probability density function to generate from, and h is a nonnegative function. Consider the following algorithm pseudo-code: Step 1. Generate Y ∼ g. Step 2. Generate E ∼ Exp(1) in the way that E = − log(U), U ∼ Unif(0, 1). Step 3. If E ≥ h(Y ), set X = Y . Otherwise go to Step 1. Step 4. Return X.This is a rejection algorithm and we want to find the density function of the generatedsamples. (d) Note that the density function f(x) in the samples is the conditional prob. f(x|accepted).Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the densityf(x) = c/x2 * e-(x^2)/2, x>1(Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)
Suppose that g is an easy probability density function to generate from, and h is a nonnegative function. Consider the following algorithm pseudo-code: Step 1. Generate Y ∼ g. Step 2. Generate E ∼ Exp(1) in the way that E = − log(U), U ∼ Unif(0, 1). Step 3. If E ≥ h(Y ), set X = Y . Otherwise go to Step 1. Step 4. Return X.This is a rejection algorithm and we want to find the density function of the generatedsamples. (a) Note that E ∼ Exp(1). What is the probability that P(E ≤ t) for any constantt > 0?(b) Given Y = x, what is the probability that Y will be accepted? (c) What is the joint probability that P(Y ≤ x, Y is accepted)? (d) Note that the density function f(x) in the samples is the conditional prob. f(x|accepted).Find f for X, subject to a constant.(e) With the results, write the pseudo-code for the densityf(x) = c/x2 * e-(x^2)/2, x>1(Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.) I have the answers for part (a) and (b) already.…
Let X have a probability density function A) Let Y = X1 + X2 + … + X25 be the sum of a random sample of size 25 from the distribution whose pdf is f(x). Approximate P[12 < Y < 21]. B) Let Yn = X1 + X2 + … + Xn be the sum of a random sample of size n from the distribution whose pdf is f(x). Discuss lim n->infinity P[-14.4 < Yn < 28.8]
5)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 a sufficient statistics for θ. Check if the statistics you have found are minimally sufficient.

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Squeezing. Let S and X have density functions satisfying b(x) ≤ fs(x) ≤ a(x) and fs(x) ≤ fx(x). Let U be uniformly distributed on [0, 1] and independent of X. Given the value X, we implement the following algorithm: if Ufx (X) > a(X), otherwise: if Ufx (X)