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

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

whosei-th element is an independent random variable having the values 0 or 1 depending on a probabilitypas 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 inZare 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 dimensionsnxnwhose 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, theijelement in the resulting matrix will be given by the binomial distribution with coeficients $\displaystyle (1, (1-p)^N)$.

I would appreciate some explanations here.