(Problem 5) Suppose we have n random variables r1, r2,..., Tn each with mean zero. How...
Let R be a commutative ring with no nonzero zero divisor and elements r1,r2,.. . ,Tn where n is a positive integer and n 2. In this problem you will sketch a proof that R is a field (a) We first show that R has a multiplicative identity. Sinee the additive identity of R is, there is a nonzero a E R. Consider the elements ari, ar2, ..., arn. These are distinct. To see O. Since R conelude that0, which...
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
Let ˜x and ˜y be zero-mean, unit variance Gaussian random
variables with correlation coefficients, . Suppose we form two new
random variables using linear transformations:
Let and be zero-mean, unit variance Gaussian random variables with correlation coefficients, p. Suppose we form two new random variables using linear transformations: Find constraints on the constants a, b, e, and d such that ù and o are inde- pendent.
Consider a sequence of random variables X1, ..., Xn, ..., where for each n, Xn~ tn. We will use Slutsky's Theorem to show that as the degrees of freedom go to infinity, the distribution converges to a standard normal. (a) Let V1, ..., Vn, ... be such that Vn ~ X2. Find the value b such that Vn/n þy b. (b) Letting U~ N(0,1), show that In = ☺ ~tn and that Tn "> N(0,1). VVn/n
11. 3 points Suppose we have ten independent random variables Xi~N1,1), X2N (21) X1N(10,1) Find the variance of the product of the random variables. In other words, find 10
11. 3 points Suppose we have ten independent random variables Xi~N1,1), X2N (21) X1N(10,1) Find the variance of the product of the random variables. In other words, find 10
Suppose that Z1 and Z2 are uncorrelated random variables with zero mean and unit variance. Consider the process defined by Yt = Z1 cos(ωt) + Z2 sin(ωt) + et where et ∼ iid N(0,σ2 e) and {et} is independent of both Z1 and Z2. Prove that {Yt} is stationary.
Suppose that X and Y are bivariate normal with density quadratic term Ξ 1 (a-2 px yty xor f(x,y) = This means that X and Y are correlated standard normal random variables since We will show that X and the new random variable Z defined as Since Z is obtained as a linear combination of normal random variables, it is also a. What is the mean of Z, call it E[Z]? b. What is the variance-covariance matrix of the random...
Problem 2: Given a collection of data { zNJS R" we define 1. The sample mean of the points is given by 2. The sample variance of the points is given by N 2 3. The covariance matrix of the points is given by Suppose that (N) S R is a collection of data points. Using Lagrange Multipliers, show that the unit vector w for which the set (i.N), where wy, has maximum variance is the normalized eigenvector of Cov(ia)...
2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean μ and variance σ2, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 7-2 log max L(μ, σ*) )-2 log ( max L(u, i) μισ (c) Explain as clearly as you can what happens to T, when our estimate of σ2 is less than 1. (d) Show...
Problem 3 Consider the linear MMSE estimator to the case where our estimation of a random variable Y is based on observations of multiple random variables, say XXX. Then, our linear MMSE estimator can be e written in the following fom: (a) Show that the optimal values of aa,a.a for the linear LMSE estimator is given as where E(X, a, Cxx is an covariance matrix of X,,X,...Xv and cxy is a cross-correlation vector, which is defined as E(x,r EtXyY (b)...