Describing the distribution of n random variables

[jimmy1]

Suppose I had n random variables, all of which have the same distribution but different mean and variances. How can I formally describe the distribution of these n random variables.

 

[Hurkyl]

If they have different means and variances, then they can't have the same distribution.

If you have n random variables all on the same set of outcomes S, then you describe that as a single random variable with outcomes in S^n. (the set of n-tuples of elements of S) And knowing the (marginal) distributions of the individual variables isn't enough: they could be dependent on each other in many different ways.

 

[jimmy1]

So for example, if I had n normally distributed random variables, and each had different mean and variance, how do I describe the distribution of the n random variables. I just need to find out what the overall mean and variance of the n random variables will be?

Would this distribution be known as a joint distribution? Would I be able to use the multivariate normal distribution??

 

[Hurkyl]

Would this distribution be known as a joint distribution?

Yes.

Would I be able to use the multivariate normal distribution??

Maybe. Maybe not.

http://en.wikipedia.org/wiki/Multivariate_normal_distribution#A_counterexample

 

[jimmy1]

Thanks for help, Hurkyl.
So, is there any general formula or method for getting the joint distribution of n random variables. For example, if all the random variables were exponential (and I knew all the dependencies of the n random variables), is there a method??
There only seems to be multivariate normal and binomial distributions??

 

[Hurkyl]

Thanks for help, Hurkyl.
So, is there any general formula or method for getting the joint distribution of n random variables. For example, if all the random variables were exponential

Let me rephrase your problem in terms of discrete distributions -- maybe it will help you understand what the problem is.

Suppose you have two random variables, each of which can take one of three outcomes.

Their joint distribution can be described by a 3x3 array of numbers -- each entry is simply the probability of that particular joint outcome.

0.05  0.20  0.10 | 0.35
0.15  0.05  0.05 | 0.25
0.10  0.15  0.15 | 0.40
-----------------+-----
0.30  0.40  0.30 | 1.00

From this joint distribution, we can also see the (marginal) distribution of the individual random variables: e.g. for one of them, the probability of each outcome is 0.35, 0.25, and 0.40. (In order)

Now, the question you appear to be asking is:

 *     *     *   | 0.35
 *     *     *   | 0.25
 *     *     *   | 0.40
-----------------+-----
0.30  0.40  0.30 | 1.00

From this diagram, can you determine the entries in the grid?



And the answer is no -- if you play around with it, you should be able to find many other joint distributions that yield this same diagram.

(and I knew all the dependencies of the n random variables), is there a method??
There only seems to be multivariate normal and binomial distributions??

What do you mean by "dependency?" If you weren't asking the question you're asking, I would have assumed that parenthetical meant you already knew the joint distribution.