5. Suppose X = (X₁, X2, X3) is a normal random vector with mean and covariancematrix Σ, where211(H)1301 0 1Find the joint distribution of Y₁ = X₁ + X2 + X3 and Y₂ = X₁ − X₂.6. Suppose X is a two-dimensional normal random vector with joint densityƒ (11₁, #₂) = k^¹ exp{-}₂ (2² + 2x² − x122–3r₁ – 202 + 4)-4)}.Find E(X) and Cov(X).μ = (2, 1, 2), Σ=

As per our guidelines we are supposed to answer only one question. 5)We have Y1=X1+X2+X3 and…

Answered: 5. Suppose X = T (X₁, X2, X3) is a…

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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Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.
Suppose Y₁,..., Yn are independent random variables each with the Pareto distribution and E(Y) = (Bo + B1Xi) Is this a generalized linear model with a canonical link? Give reasons for your answer.
Let Z₁ and Z₂ be independent standard normal random variables. Let pe [-1, 1]. Find a matrix L such that has a distribution. X = LZ N (1). (1))
(a) Let X, Y be independent random variables with respective PMFS Px(x) = (-) )* > Py (Y) = (-) )", y=1, 2, 3, . . . .2 , х-0, 1, 2, 3, . . Find P(X+Y=3)
2
Let Y₁, Y2, Y3, Y₁, Y5 be a random sample of size 5 from a standard normal population. Find the moment generating function of the statistic: X = 2Y₁² + Y₂² + 3Y3²+ Y₂² +4Y5² Let Y₁, Y₂, Y3, Y4, Y5 and X₁, X2,..., X, be independent and normally distributed random samples from populations with means ₁ = 2 and ₂ = 8 and variances ₁² = 5 and ₂² = k₁ respectively. Suppose that P(X-Y> 10) = 0.02275, find the value of 0₂² = k.
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.
3. If X;~N(0,1), i = 1,2,3,4 are independent Normal random variables and Y is given as Y =(X,+X,)' +(X, + X,)' . If aY~x², . find the constant a. 2
Is the following statements correct or incorrect: Suppose that a random vector Y' = (Y1, Y2, Y3) has multi-normal distribution with mean vector u and variance-covariance matrix £ given by and Σ 흑 0 1 respectively. CH), Then the distribution of Z = Y is N 1
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Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.
2. (a) When two random variables (X, Y) follow a bivariate normal distribution, the covariance matrix Σ is defined as Σ ρσισ2 (010₂ Posts), - = where p is the correlation, o, o are the variances of X and Y respectively. Use mvrnorm from the MASS library (use library(MASS)) to generate 100 samples from a bivariate normal Call the first column x distribution with μ = (₁, ₂) with μ₁ = (11). 2,42 = 3, and Σ = = and the second column y. (b) Plot the histogram of x and overlay it with the corresponding marginal normal density. Repeat for y. (Recall the marginal distribution of X is N (μ₁, 0²).) (c) Produce a scatter plot of x and y (use plot). Compute the sample correlation coefficient (use cor) and compare with the population correlation p. (First work out p in the Σ given.)

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5. Suppose X = T (X₁, X2, X3) is a normal random vector with mean and covariance matrix X, where 2 1 30 1 0 1 Find the joint distribution of Y₁ = X₁ + X₂ + X3 and Y₂ = X₁ - X2. 1 1 μ = (2,1,2)T, Σ