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If Xp×1∼Np(μ,Σ)X_{p\times1}\sim N_p(\mu,\Sigma)Xp×1​∼Np​(μ,Σ)​​, then AX follows, where A is a matrix of rank q≤p,\leq p,≤p,​​​
Question

If Xp×1Np(μ,Σ)X_{p\times1}\sim N_p(\mu,\Sigma)​​, then AX follows, where A is a matrix of rank qp,\leq p,​​​

A.

Np(Aμ,AΣA)N_p(A\mu,A\Sigma A')​​

B.

Nq(Aμ,AΣA)N_q(A\mu,A\Sigma A')​​

C.

Np(Aμ,AΣA)N_p(A\mu,A'\Sigma A)​​

D.

Nq(Aμ,AΣA)N_q(A\mu,A'\Sigma A)​​

Correct option is B

Given:AμA\mu
XNp(μ,Σ)X\sim N_p(\mu,\Sigma)​​​
A is a q×pq\times p​​ matrix of rank q.
Formula Used:
If XNp(μ,Σ)X\sim N_p(\mu,\Sigma).​, then
AXNq(Aμ,AΣA).AX\sim N_q(A\mu,A\Sigma A').
Solution:
Since A has q rows, AX is a q-dimensional random vector.
Its mean is  and its covariance matrix is AΣA.A\Sigma A'.​​​
Hence,
AXNq(Aμ,AΣA).AX\sim N_q(A\mu,A\Sigma A').​​​
The correct answer is (b) Nq(Aμ,AΣA). N_q(A\mu,A\Sigma A').

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