Pls help with (a), (b), (c), (d) 3. Suppose that g is an easy probability density function to generate from, and h is a non-negative function. Take a close look at the following algorithm pseudo-code:Step 1. Generate Y ~g.Step 2. Generate E~ Erp(1) in the way that E = – log(U), U~ Unif(0, 1).Step 3. If E 2 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 - Erp(1). What is the probability that P(E < t) for any constantt> 0?(b) Given Y = 1, what is the probability that Y will be accepted?(e) What is the joint probability that P(Y < 1,Y is accepted)?(d) Note that the density function f(z) in the samples is the conditional prob. f(z|accepted).Find f for X, subject to a constant.

Given g is an easy probability density function to generate from and h is an non-negative function.

Answered: Suppose that g is an easy probability…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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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.…
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]
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.)
Theorem 6.4 states that the moment-generating function of the gamma distribution is given by Mx(t) = (1-βt)^(-α).
Take the model y = y x'1β1+x'2β2+ϵ with E(xe)=0 Suppose B1 is estimated by regressing Y on X1 only. Find the probability limit of this estimator.
Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0
Suppose that X and Y have a joint probability density function given by: ( only for D) and E), ignored A),B)and C))
Verify that p(x) = 3x−4 is a probability density function on [1,∞) and calculate its mean value.
Let p(x) = cx^2 for the integers 1, 2, and 3 and 0 otherwise. What value must c be in order for p(x) to be a legitimate probability mass function?
Suppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?
Verify that p(x) = 3x - 4 is a probability density function on [1, oo)and calculate its mean value.
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)

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