4.Suppose that X, Y, and Z are multivariate Gaussian random variables with zero means andthe covariance matrixoXY OXZ31-1o oyz11%3DOZXOZY-1 1(a) Given that Y = 1 and Z = -1, find an MMSE estimate of X. Compute the conditional MSE ofthis MMSE estimate.(b) Given that Y +2Z = 2, find an MMSE estimate of X. Compute the conditional MSE of this%3DMMSE estimate.

In this problem, we are given three multivariate Gaussian random variables, X, Y, and Z, with zero…

Answered: 4. Suppose that X, Y, and Z are…

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 the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxy
In a Bernoulli process with success rate .4, if we consider the total number of successes X which happened on trials 7,9,15,16, 23, and if Y is the total number of successes on trials 15,23, 32,38, 43, then what is the covariance of X with Y?
1) Assuming you have a data matrix X that has n rows and p variables and you know both µ and Σ. How is (X- µ)‘Σ-1(X- µ) distributed? 2) Assuming that you don’t know the values of µ and Σ. How is the statistical distance distributed as n-p gets large?
Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variables
Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?
Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
Given three independent Bernoulli processes Xn, Yn, Zn with success probabilities given as 0.10, 0.87, and 0.26, respectively. X and Y are merged into one Bernoulli process which is merged with Z after. Find the resulting success probability of the last Bernoulli process. (Note: the answer should be in 4 decimal points)
In the simple linear regression model , the Gauss Markov ( classical ) assumptions guarantee that the OLS estimator of the unknown parameters is BLUE. Among those assumptions , in order to have consistency of the OLS estimator we need ( this is a question on the necessary condition ): Question 4Select one: a. we need that the errors are not correlated with each others and that they have zero mean b. we need that the errors are homoscedastic ( all have the same variance) c. we need that the residuals are not correlated with the explanatory variables and that the residuals have zero mean d. we need that the errors have zero mean and that they are not correlated with the regressors ( no endogeneity) e. we need that the errors are homoscedastic ( all have the same variance) and not correlated with each others ( no serial correlation)
A random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance of Y if only 2.5% of its elements is a subset of X.
Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of X1/(X1+X2+X3+X4) and (X1+X2)/(X1+X2+X3+X4) and (X1+X2+X3)/(X1+X2+X3+X4) using Jacobian method?
If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?

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