## Question 1. (20 points)

Consider a 2 Gaussian random vectorx = (X,X,)T and a matrix A∈3×2; assume that with . Introduce = (Z1 , Z2 , Z3)T = Ax.

(a) Give the condition on for the matrix ∑to be a covariance matrix. 

(b) For which value of do we have X1 ⟂⟂ X2 ? 

Assume that c is such that the matrix ∑x is a covariance matrix.

(c) Compute E(z) and var(z), and give the distribution of z. 

Hint: Compute each entry of the vector E(z) and of the matrix var(z); note that var(z) should  depend on c.

Consider the vector v = (1,−1, 1); note that ATv = 0.

(d) Use the vector v to defifine a principal component of z, and give the variance of this principal component. 

(e) What is the fraction of variance explained by the fifirst two principal components of the random vector z? For which values of c does the fifirst principal component of the random vector z explain 100% of its variance? 

Hint: You may consider det(∑x).

(f) Give an expression for Z3 in terms of Z1 and Z2 . 

## Question 2. (11 points)

Consider a centered Gaussian process (Zx)xwith covariance kernel K(x, x′ ) = e−|xx′|, for x and x′ ∈ .

Assume that the realisation Z0 = z0 was observed, with z0 ∈ (i.e. observation at x = 0). Consider x and x′ ∈ .

(a) Give the distribution of the random variable Zx|Z0=z0 . Draw a scheme illustrating the appearance of the map x E (Zx|Z0 = z0) , withx . 

(b) Compute c-cov (Zx , Zx′ | |Z0 = z0) ; what can be concluded forx < 0 and x> 0? 

## Question 3. (21 points)

Consider two independent random samples x1, , xnx i.i.d. x ∼ Nd(Ux, ∑), and y1, , yny i.i.d.y ∼ Nd(uy, ∑). Note that these two samples are defifined from Gaussian distributions withsamecovariance matrix, but with potentially difffferent means; the sample related to x is of size nx, and theone related to y is of size ny (and the random vectors xi and yi are independent for alli ∈ {1, , nx}and j∈ {1, , ny}). Let {̄x, Sx} and {̄y, Sy} be the corresponding sample mean and corrected (i.e.unbiased) sample covariance estimators. Consider the pooled variance estimator EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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