Let say a value N >> 1 is distributed among m>1 members, which can follow some distribution (Binomial or Gaussian). Is there any generic term to represent the multiplication of the distribution. For example, a silly guess for multiplication of uniform distribution will be (N/m)^m. Is there any generic term for other distributions too?
I do not really follow what you mean with N, m and multiplication of a distribution (multiplication of two distributions makes more sense) but if X and Y are independent and Gaussian with expectation 0 and standard deviations s_x and s_y, then Z=XY has the probability density function K_0(|z|/(s_x s_y))/(pi s_x s_y), where K_0 is the zero order modified Bessel function of the second kind.
The question is not clear. Do you looking for the joint probability function of two or more variables, where each has a certain type of distribution (Gaussian for example)?
Let there are m numbers [n1,n2,n3...nm], The sum of these numbers is N, i.e., N =n1+n2+...nm.
What would be the product of these numbers i.e., P=n1xn2x...nm.
These number can take any values, one case is when all of them are equal i.e., n1=n2=n3..=nm=n. In that case the product P=n^m. I suppose that this case is called uniform distribution i.e every number is uniformly distributed.
I thought if these number are distributed differently like Gaussian or some thing and there is some expression for them as well. I think usually the distribution terms are used for probabilities, but the concept is same, i.e., in probabilities the sum is 1 here the sum is N.
I think the clue is to use the characteristic function of N. Suppose now, N = x1+..+xn. Where {xi} is independent identical random variable with Gaussian distribution. Then The characteristic function of N can be computed from E(exp(i t N)). You can also compute the distribution of P = x1*x2*...*xm in a similar way. If you get the characteristic function of N or P, then a Fourier transform of it will be the desired probability density function.
For more information, see http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29.
Cheers.
Je ne me juge pas compétent pour répondre à cette question.
Cordialement
- Badges
- Science topic