# Help in some problems in probability (urgency)

• Mar 16th 2010, 05:14 PM
assuom
Help in some problems in probability (urgency)
Good morning every body
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.
• Mar 16th 2010, 05:40 PM
Anonymous1
Why not post one at a time, or each part successively. I want to help, but I just tried to read this and became terminally bored by the second sentence.
• Mar 16th 2010, 08:15 PM
assuom