# Thread: Help in some problems in probability (urgency)

1. ## 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.

2. 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.

3. ## sorry about that

sorry about that, but if you know a solution of 1 problem that is good, and send me it
thanks

4. Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. It is better for forum organization and better for you to get your questions answered in a more timely manner if you start new threads as necessary for remaining questions. eg. If you have five questions, post two of them in two threads and start a new thread for the remaining one etc.

And if the question has more than two parts to it, it is best to post only that question and its parts in the thread and start a new thread for other questions.