2. [Transformations] Consider X ~ U(0, 1) with the probability mass function1, x = (0,1)fx(x) =0, otherwise(a) Find support and PDF for Y = log(X).(b) Find the moment generating function (MGF) of Y.(c) Using MGF, find the first two moments of Y.

Part (a):To find the support and probability density function (PDF) for Y=log(X), we need to first…

Answered: 2. [Transformations] Consider X ~ U(0,…

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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70. Let X~ Xn. (a) Find the moment generating function mx (t) for X. (b) Use the mgf for X to derive the formula for the mean of X, μ = (c) Use the mgf for X to derive the formula for the variance of X, o² = V(X) = E(X²) - [E(X)]² = m (0) - [m'x (0)]². E(X) = m'x (0).
Example: The Gamma Distribution Suppose X ~ T(a, B), then M(t) = dx BaT(a) 22 Now we do a little high school algebra. Define & = Then one may write 1-Bt" Equation (22) as: M(t) = ()ª / (23) dx One recognizes the integrand of Equation (23) as a r(a, 6) density, hence 8. M(t) = (Gr"(1) = (1 - AY• (24) (1 – Bt)a Exercise 5. Verify the algebra leading from Equation (22) to Equation (23).
Consider the log likelihood function L(x | 0) where 0 = (01,02)' is a vector of parameters . Let 0* be the value of 0 that makes the gradient of L(x | 0) with respect to 0 equal to vector 0. The hessian matrix evaluated at V3 The log likelihood at 0* is then 1 2 critical 0* is given by H = V3 (a) a local maximum (b) a local minimum (c) a saddle point (d) 0
1)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).
Exercise 6. Using the MGF given by Equation (24), deduce the expressions for the mean and variance we found earlier. Exercise 7. Prove the algebraic identity: 2g2 + xt = + ut 25 2g2
Logistic regression aims to train the parameters from the training set D = g(0¹ x) {(x(i),y(¹)), i = 1, 2,...,m, y € {0, 1}} so that the hypothesis function h(x) 1 (here g(z) is the logistic or sigmod function g(z) 1+ e-z new instance x being labeled as 1. Please derive the following stochastic gradient ascent update rule for a logistic regression problem. 0₁ = 0₁ + α(y(¹) - h₂(x)))x) = ) can predict the probability of a
(vi) Given below is a joint pdf of X₁ and X2₂ which are independent exponential random variables with parameter X. f(x1, x2) Find the joint probability density function (pdf) of Y₁ = X₁ − X₂ and Y₂ = X₁ + X₂ using the transformation method. =e-21-x2 0 < x1 <∞0,0 < x₂ <∞
Let X and X, independence r.v. have same moment generating function (-) where tsa, let Y=aX+bX, , then My (1) is 1
Calculate correlation and probable error between physical parameters (temperature and humidity) Number of families(x)=2 Number of sites(n)= 1 Temperature(y) in autumn= 24.1 Humidity(y) in autumn= 35 20 1 19 Calculate correlation and probable error between physical parameters (temperature and humidity) Using the following formula coefficient of correlation (r). r = n([xy)-([x) (Ey) V[n(Ex 2)-(Ex) 2][n(Ey2)-([y) 2] I

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2. [Transformations] Consider X ~ U(0, 1) with the probability mass function 1, x = (0,1) fx(x) = 0, otherwise (a) Find support and PDF for Y = log(X). (b) Find the moment generating function (MGF) of Y. (c) Using MGF, find the first two moments of Y.