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I have two questions:

Let A~N(mu,sigma), B~U(0,θ) and C=A*B

1.What is the distribution of the product of a normal and uniform matrices, C? (How we can prove it)

2.What statistical properties of these two matrices are preserved in their product? (I understand, for example, if both matrices have normal distribution, their product will preserves their euclidean distance and their inner product, but that s not the case when one matrix is uniform).

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Do you mean what is the distribution of the product of these two random variables? What is the connection with matrices? –  André Nicolas Jun 24 '13 at 6:29
    
Yes I am asking about the distribution of the product of these two matrices, –  user1468089 Jun 24 '13 at 7:12
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They are independent –  user1468089 Jun 24 '13 at 7:13
    
What do you mean by "their product will preserves their euclidean distance and their inner product"? Euclidean distance from what to what? Inner product of what? –  Robert Israel Jun 24 '13 at 12:40
    
Sorry for my unclear question, I meant in the product of C=A*B, what statistical relationship of A and B elements are preserved?, e.g., if these two matrices have normal distribution the euclidean distance between A elements would be preserved in the resulted matrix (C) (this is also the same for B), but if A and B have different distributions (in my case normal and uniform), the euclidean distance will not be preserved, and I am wondering what statistical relationship between the elements is maintained and how can I show that. –  user1468089 Jun 24 '13 at 22:59
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1 Answer 1

The multivariate case (if that's why you're talking about matrices) looks rather complicated, but let's look at the univariate case.

According to Maple, $C = AB$ has moment generating function $$M_C(t) = E[e^{tAB}] = \frac{\sqrt {2\pi }}{2 \theta \sigma t} \left( {\rm erfi} \left( \frac{\sigma^2 t \theta + \mu}{\sqrt{2}\sigma}\right) - {\rm erfi} \left(\frac{\mu}{\sqrt{2}\sigma}\right) \right) { {\rm e}^{-\mu^2/(2 \sigma^2)}} $$

I doubt that this is a "named" distribution.

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Thanks Robert for your reply, could you please advise me if you know any reliable resources for further reading, especially about my second question. –  user1468089 Jun 24 '13 at 12:18
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