6. Let X1 and X2 be independent and identically distributed random variables with joint mgfMx,x, (t1, t2) = [(1 – t,)(1 – t2)]¯1.%3D6.1 Give the marginal mgf of X2, mx, (t2).6.2 Give the mgf of Y = X, + X2, my(t).

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Answered: 6. Let X1 and X2 be independent and…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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3.7. Consider the performance function Y = 3x1-2x2 where Xi and X2 are both normally distributed random variables with Ax' = 16.6 0% 2.45 μΧ2 = 18.8 ơx.-2.83 The two variables are correlated, and the covariance is equal to 2.0. Determine the probability of failure if failure is defined as the state when Y 0 3.8. The resistance (or capacity) R of a member is to be modeled using R = R,MPF where Rn is the nominal value of the capacity determined using code procedures and M, P, and Fare random variables that account for various uncertainties in the capacity. If M, P, and F are all lognormal random variables, determine the mean and variance of R in terms of the means and variances of M, P, and F.
Question 2. Show that, for a simple linear regression, R² = ry, where R² is the coefficient of determination and rxy is the sample correlation between Y and X. Hint: Σ" (Υ - Y)2 Σ1(Y₁-Y)²¹ SxY √SXXSYY' where Sxy = 1 (X; — X)(Y; — Y), Sxx = ₁1(X; - X)², Syy = ₁₁(Y₁ - Y)². R² = "XY =
Given (x + 2y) ,0
(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)
Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1
Page 263, 4.4.8
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.
7. Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: The marginal function of x %3D
18 Given o = (aX + 3Y) where X and Y are zero - mean random variables with variances o =4 and o= 16. Their correlation coefficient is p=- 0.5 %3| (a) Find a value for the parameter 'a' that minimizes the mean value of w (b) Find the minimum mean value.
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If pdf of a random variable is given by fx(x) = e* for x20 find My(v), m, and m,. X

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6. Let X1 and X2 be independent and identically distributed random variables with joint mgf Mx,x, (t1, t2) = [(1 – t,)(1 – t2)]¯1. %3D 6.1 Give the marginal mgf of X2, mx, (t2). 6.2 Give the mgf of Y = X, + X2, my(t).