I have a definition problem that I would like to understand. I wondered whether somebody could give me some help or recommend me a good reference.
Assume I have a vector
whose i-th element is an independent random variable having the values 0 or 1 depending on a probability p as follows.
My notation until here, I hope it's understandable.
The publication that I'm reading claims that the elements in Z are independent random variables (I understand that). Now, if we obtain the outer product of that vector by multiplying the vector with its transpose, we obtain a matrix of dimensions nxn whose elements are not independent:
Basically my question is; is not the product of two independent random values simply another independent random value? I do not understand why the matrix has now values which are not independent.
Moreover, the publication suggest that they become independent if we fix a value:
and that this N scalar results from a binomial distribution with coefficients . Then, the ij element in the resulting matrix will be given by the binomial distribution with coeficients .
I would appreciate some explanations here.