A random point in space is given by three rectangularcoordinates forming a system of normal random variables with the probabilitydensity√316773/2 exp{−1 12x² + 4y² − 2y(z + 5) + (z + 5)²]}.f(x, y, z)(a) Find the covariance matrix, (b) determine the locus of points when theprobability is 0.01.

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Answered: A random point in space is given by…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Let W = 2X + 5Y – 3Z, where the random variables X, Y, and Z satisfy E(X)= 2, V(X)= 5, E(Y)= -3, V(Y)=4, E(Z)=7, V(Z) = 6, Cov(X,Y) = -1, Cov(Y, Z) =3, Cov(Z, X)= 2. Calculate E(W), V (W) and Cov(W, Y).
Find E(R) and V (R) for a random variable R whose moment-generating function ismR(t) = e2t(1-3t2)-1
A stochastic process (SP) X(t) is given byX(t) = Asin(ωt + Φ)where A and Φ are independent random variables and Φ is uniformly distributed between 0 and 2π.a) Calculate mean E[X(t)]. b) Calculate the auto-correlation RX (t1,t2).c) Is X(t) wide sense stationary (WSS)? Justify your answer.Now consider that X(t) is a Gaussian SP with mean μX (t) = 0.5 and auto-correlation RX (t1,t2) =10e−14 |t1−t2|. Let Z = X(5) and W = X(9) be the two random variables.d) Calculate var(Z), var(W), and var(Z + W). e) Calculate cov(ZW).
Let (X₁, X₂) be a pair of standard exponential random variables with the countermonotonicity copula and let U be a standard uniform random variable. a) Find a stochastic representation of the form (X₁, X₂) = (f(U), g(U)) for functions f and g which shows how (X₁, X2) can be simulated. b) With the help of a), use software and numerical integration to calculate the minimal correlation for two standard exponential random variables.
A Random process X(t) is applied to a network with impulse response h(t) = u(t) exp (-bt) where b > 0 is a constant. The cross- correlation of X(t) with the output Y(t) is known to have the same form, Ryy(t) = u(t)t exp(-bt) (i) Find the auto-correlation of Y(t).
Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If B is the linear minimum variance estimate of B based on y, show that the covariance of the error B - B has rank n - m.
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
For a GLM with canonical link function, explain how the likelihood equations imply that the residual vector e = (y – @) is orthogonal with C(X).
(b) Compute the covariance matrix of the data given by x1 = [2 1]', x2 = [3 2]', x3 = [2 3]'and x4 = [1 4]'.
we have the orthogonal matrix of U from (a), please calculate the (b) thanks :)

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