Hi everybody,

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

$\displaystyle \mathbf{Z} \in \Re^n$

whose i-th element is an independent random variable having the values 0 or 1 depending on a probability p as follows.

$\displaystyle P(Z_i) = 1 \text{ if } Z_i > p \text{, 0 elsewhere}$

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:

$\displaystyle J = \mathbf{Z} \cdot \mathbf{Z}^T$

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:

$\displaystyle N = \sum Z_i$

and that this N scalar results from a binomial distribution with coefficients $\displaystyle (1, p)$. Then, the ij element in the resulting matrix will be given by the binomial distribution with coeficients $\displaystyle (1, (1-p)^N)$.

I would appreciate some explanations here.