Help in some problems in probability (urgency)

Good morning every body

Please help me as soon as possible and thanks’ a lot

1-9: hypothesis testing with correlated noise: consider the hypothesis

H0 = 0 and H1 different of 0

And the observations: zi= θ + wi i = 1,…,n

With wi zero-mean jointly Gaussian but not independent. Denoting w = [ w1 … wn]’

One has the covariance matrix (assumed given) E[w*w’] = P

For the above:

1- Specify the optimal hypothesis test for false alarm probability α

2- Solve explicitly for n=2, P = [1 05; 0.5 1] and α = 1%.

1-10: Partial derivative with respect to a matrix: the partial derivative of a scalar q with respect to the matrix A = [aij] is defined as [∂q/∂A] = [∂q/∂aij]

Prove that:

1- For B symmetric , [∂tr[ABA’]/∂A] = 2AB

2- For B not symmetric, , [∂tr[AB]/∂A] = B’

1-13: conditional probability density function (pdf) of the sum of two Gaussian random variables:

If x1 and x2 independent, the pdf of their sum x=x1+x2 conditional on x1 is

p(x\x1) = px2(x-x1) = N(x;x1+x-2, P22) where: p([x1 x2]’) = N{ [x1 x2]’ ; [x-1 x-2]’, [ P11 P12;P21 P22]

Then, Find the pdf in general case?

1-14: probability matrix: we have P{x (k) = xj\x (k-1) = xi} = πij i,j = 1, … , n

Find ∑j=1n (πij)

1-16: moments of a quadratic form with non-zero-mean random variables:

Consider the random variables x and y with means x- and y-, respectively, and with covariance Pxx, Pyy, Pxy.

Evaluate E[x’Ay]

please if anyone know the solution of all or one problem send for me the solution.

And thanks again for your help.