6. (For stat 5504 only). Let Y = X² +XZ where X and Z are independent, normallydistributed and E(Z) = E(X)= 2 and E(Z²) = E(X?) = 3,(a)if X = 1.8?(b)Y = aX +b (find a, b). What is this value if X = 1.8?Find the best (among all) predictor of Y from X. What is this valueWhat if we only want the best linear predictor of Y from X. I.e.

Given :E(Z)=E(X)=2 and E(Z2)=E(X2)=3a) The predictor of Y from X. if X=1.8.b) The best linear…

Answered: 6. (For stat 5504 only). Let Y = X² +XZ…

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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Consider randomly selecting n segments of pipe and determining the corrosion loss (mm) in the wall thickness for each one. Denote these corrosion losses by Y₁' Yn. The article "A Probabilistic Model for a Gas Explosion Due to Leakages in the Grey Cast Iron Gas Mains"+ proposes a linear corrosion model: Y; = t;R, where t; is the age of the pipe and R, the corrosion rate, is exponentially distributed with parameter 1. Obtain the maximum likelihood estimator of the exponential parameter (the resulting mle appears in the cited article). [Hint: If c> 0 and X has an exponential distribution, so does cX.] O Â = O Â = O λ = * * O n Srit) i = 1 n j = 1 Σ i = 1 n n i = 1 n LY i = 1 n 0 1 = L (rt) 2 i = 1 | = 1 n n Y; | = 1
Imagine you have two estimators X and Y, for unknown parameter µ. Assume that they are estimated from independent datasets. Let Z = }(X+Y) be the average of these two estimators. To write the symbol µ, you can write mu. For other math notation, you can E[X], Var(X) and (X+Y)/2 or (1/2)(X+Y). Part a - If X and Y are unbiased estimators, E[X] = E[Y] = µ, then is Z unbiased? Explain why or why not. Your explanation should include a short derivation. Part bL Assume Var(X) = Var(Y). Is Z a lower variance or higher variance estimator than X? Explain your answer. Your explanation should include a short derivation.
Show that if t" (t + t) where t and ť are both most efficient estimators with variance v, then var (t") = v (1 + p).
Suppose X and Y are two independent variables with variance 1. Let Z = X+bY where b > 0. If Cor(Z, Y ) = 1/2, what is the value of b?
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6. Find the mean and variance of fx (x) = 2e eA* la.2...) (x) = de
The functions i)= e **2f,(x) = e*-3 %3D are O A. Insufficient information to conclude None of them O B. O C. Neither linearly dependent or independent OD. Linearly independent O E. Linearly dependent
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?.
Let X ~ F(a, ß). If E(X)= % and E(X²)= a+a² If â = nX? find the method of (n-1)s² | .3 moments estimator of
Please answer number2
If 0 is the angle between the two regression lines. Show that sin 0 < 1 - r? (r is the correlation coefficient )
5.

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