If a and b are two random vectors distributed with some Probability density function. Then what would be the probability density function of following:
1. Random matrix obtained as A = diag(a),
2. Trace of a matrix, Tr(A)
3. Norm of a vector or Matrix,
4. Schur-Hadamard product two random vectors,
5. Outer product of two random vectors
Please refer to an Image attached here in shows problem with which I'm dealing. Would appreciate if anyone can point even me to some links for reference and books.
Thanks
The word random implies unpredictable, but does not specify the distribution. There must be a dozen ways to select a vector "at random" each with a different distribution. To illustrate with a simpler example, suppose we select a random chord in a circle of radius 1. What is the probability that the length exceeds sqrt(3)? If we find our chord by choosing two points uniformly on the circle, the answer is 1/3. But we could determine our chord by selecting one point uniformly in the interior of the circle, and then drawing the unique radius through this point, and then constructing the perpendicular line through our point to the radius. This gives the unique chord having our point as its midpoint. The probability that this is longer than sort(3) is 1/4. Or a third method: Select the radius at random, select a point on the radius uniformly, and then construct the perpendicular. Now the probability is 1/2.
The mechanism chosen for the random vector selection determines the distribution.
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