Question

Review: Manipulating Multivariate Gaussians 1 point possible (graded) Recall that a multivariate Gaussian N(μ⃗ ,Σ) is...

Review: Manipulating Multivariate Gaussians

1 point possible (graded)

Recall that a multivariate Gaussian N(μ⃗ ,Σ) is a random vector Z=[Z(1),…,Z(n)]T where Z(1),…,Z(n) are jointly Gaussian , meaning that the density of Z is given by the joint pdf

f:Rn → R
Z ↦ 1(2π)n/2det(Σ)−−−−−−√exp(−12(Z−μ⃗ )TΣ−1(Z−μ⃗ ))

where

μ⃗ i =E[Z(i)],(vector mean).
Σij =Cov(Z(i),Z(j))(positive definite covariance matrix).

Suppose that Z∼N(0,Σ). Let M denote an n×n matrix.

What is the distribution of MZ?

Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Consider a multivariate random sample X1, . . . , Xnwhich comes from p-dimensional multivariate distribution...
Consider a multivariate random sample X1, . . . , Xnwhich comes from p-dimensional multivariate distribution N(µ, Σ), with mean vector µ ∈ R p and the variance-covariance positive definite matrix Σ. Find the distribution and its parameters for the matrix nXTHX where H is the idempotent centering matrix H = In −1/n (1n ⊗ 1n ) and 1n is the n-dimensional vector of all 1’s.
Confidence interval Concept Check 1 point possible (graded) As in the previous section, let X1,…,Xn∼iidexp(λ). Let...
Confidence interval Concept Check 1 point possible (graded) As in the previous section, let X1,…,Xn∼iidexp(λ). Let λˆn:=n∑ni=1Xi denote an estimator for λ. We know by now that λˆn is a consistent and asymptotically normal estimator for λ. Recall qα/2 denote the 1−α/2 quantile of a standard Gaussian. By the Delta method: λ∈[λˆn−qα/2λn−−√,λˆn+qα/2λn−−√]=:I with probability 1−α. However, I is still not a confidence interval for λ. Why is this the case?
Review: Central Limit Theorem 1 point possible (graded) The Central Limit Theorem states that if X1,…,Xn...
Review: Central Limit Theorem 1 point possible (graded) The Central Limit Theorem states that if X1,…,Xn are i.i.d. and E[X1]=μ<∞ ; Var(X1)=σ2<∞, then n−−√[(1n∑i=1nXi)−μ]−→−−n→∞(d)Wwhere W∼N(0,?). What is Var(W)? (Express your answer in terms of n, μ and σ).
Method of Moments Concept Question II 1 point possible (graded) Let (E,{Pθ}θ∈Θ) denote a statistical model...
Method of Moments Concept Question II 1 point possible (graded) Let (E,{Pθ}θ∈Θ) denote a statistical model associated to a statistical experiment X1,…,Xn∼iidPθ∗ where θ∗∈Θ is the true parameter. Assume that Θ⊂Rd for some d≥1. Let mk(θ):=E[Xk] where X∼Pθ. mk(θ) is referred to as the k-th moment of Pθ . Also define the moments map: ψ:Θ →Rd θ ↦(m1(θ),m2(θ),…,md(θ)). What conditions on ψ do we have to assume so that the method of moments produces a consistent and asymptotically normal estimator?...
Comparing Quantiles 1 point possible (graded) Let Z∼N(0,1). Then Z2∼χ21. The quantile qα(χ21) of the χ21−distibution...
Comparing Quantiles 1 point possible (graded) Let Z∼N(0,1). Then Z2∼χ21. The quantile qα(χ21) of the χ21−distibution is the number such that P(Z2>qα(χ21))=α. Find the quantiles of the χ21 distribution in terms of the quantiles of the normal distribution. That is, write qα(χ21) in terms of qα′(N(0,1)) where α′ is a function of α. (Enter q(alpha) for the quantile qα(N(0,1)) of the normal distribution.) qα(χ21)=
1. Given β = XT 1×nAn×nXn×1, show that the gradient of β with respect to X...
1. Given β = XT 1×nAn×nXn×1, show that the gradient of β with respect to X has the following form: ∇β = X T (A + A T ). Also, simplify the above result when A is symmetric. (Hint: β can be written as Pn j=1 Pn i=1 aijxixj ). 2. In this problem, we consider a probabilistic view of linear regression y (i) = θ T x (i)+ (i) , i = 1, . . . , n, which...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT