2. Prove the following properties of expectation and variance:If a and b are constants, X and Y are random variables, thenE(aX+b)=aE(x)+bvar (aX+ b) = a? var (X)var (X+ Y)= var (X) + var (Y) + 2 cov(X, Y)=var (X) + var (Y) + 2po;0yOf which p is correlation coefficient, oz and o, are standard error of Xand Y.3. Assume that X- N(6, 4). What is the probability that 2 10?

Answered: 2. Prove the following properties of…

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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2. A random sample X₁, X2, X3 of size 3 is drawn from a population with mean and variance o2. Let 1 3 Đi (Xi+X+X3) - (X₁ + X2 + X3), (3X₁ — X₂ +2X3). - 4 (a) Show that ₁ and ₂ are both unbiased estimators of the population 2 = mean . (b) Compute the variance of ₁ and 2. (c) Which estimator would you use? Explain your answer.
3) A random variable X takes the values 0, 2 and 3 with probabilities 0.3, 0.1 and 0.6, respectively. A random variable in the form of Y = 3(X-1)2 is defined. a) Find the expected value and variance of the random variable X. b) Find the variance of the Y random variable. c) Find the cumulative distribution function of the Y random variable.
A coin is tossed twice. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find (a) P( W=2 | Z= 1) (b) P( Z=0 | W = 1) ; (c) are Z and W independent?
3. Let X1, X2, ..., Xn be a random sample from the population N(u, o²) and Y1, Y2,...,Ym a random sample from the population N(uy, o²), where both means (ua and uy) are assumed known. Note that the two distributions have a common variance of o?. (a) If the X's and Y's are independent, show that the MLE for the common variance is: E(Xi - Ha)² +E(Y; - Hy)² n + m (b) Is the MLE found above unbiased for o??
If X and Y are independent random variables with E(X) = 10, Var(X) = 4, E(Y) = 12 and VAR(Y) = 3, If W =-2X +Y Find E(W) and STD (W) (a) (b) If X and Y are not independent, with E(X) =10, VAR(X) = 5, E(I) =18, VAR(Y) = 3 and CovX,Y) = -2. If W = X – 2Y Find E(W) and STD(W)
Suppose the random variables X,, X,,X3 have the population variance/covariance matrix: -2 0 E=-2 5 1 Calculate the principal components and interpret their meaning. What can be gained from knowing these values?
2. Some properties of Expected value and variance of a random variable. a) Assume that X is an arbitrary discrete random variable, and a and b are constant. Using the definitic Show: and E(aX + b) = a · E(X) + b V(aX + b) = a² · V(X ) Stat 3128 Ali Mahzarnia P STAT 3128 Ali Mahzarnia b) Justify the computational formula of Variance of a random variable which is to justify : V(X) = E[(X – µ°] = Ex – µ)P • ptx) = | 2: Here needs justification By Cauchy Schwarz inequality it can be shown that the right hand side is always positive. Analogs expression in Mechanic : Parallel axis theorem Iem = I– md² Moment of Moment of inetria about Inertia of an an axis shifted object about the center of a mass by d from center of mass (a parallel shift) Icm is dispersion around the mean and is like second central moment (variance) I is like second moment if d is mean m is like sum total all the weight of each of the x which all add up to 1 d squared is like squared of mean since we,
Suppose that Z is a discrete random variable with positive expectation. If Var(Z) = 3 and E(Z²) = 4, match up the following quantities. E(1 - 4Z) Drag answer here E(Z² - 5) Drag answer here Var(Z² – 5) Drag answer here Var(2Z) Drag answer here Var(2 - Z) Drag answer here 3 Cannot be determine d 12 -1 -3
Given that X and Y are independent random variables and E(X) = 4, E(Y) = 7, Var(X) = 3, Var(Y) = 6; find Var(8X - 2Y)
Prove the following

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