Describing the distribution of n random variables

In summary: In that case, there isn't a specific method, but instead you would need to find the marginal distributions of the individual variables and use those to calculate the joint distribution.
  • #1
jimmy1
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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.
 
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  • #2
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.
 
  • #3
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??
 
  • #5
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??
 
Last edited:
  • #6
jimmy1 said:
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.

Code:
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:



Code:
 *     *     *   | 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.
 

What is meant by "describing the distribution" of n random variables?

Describing the distribution of n random variables refers to the process of quantifying and understanding the patterns and characteristics of a set of random variables. This typically involves calculating measures such as mean, median, and standard deviation, as well as visualizing the data through graphs or charts.

Why is it important to describe the distribution of n random variables?

Describing the distribution of n random variables allows us to gain insights into the underlying patterns and relationships within the data. This information can be used to make predictions, identify outliers, and make informed decisions based on the data.

What are some common methods used for describing the distribution of n random variables?

Some common methods for describing the distribution of n random variables include calculating measures of central tendency (such as mean, median, and mode), measures of variability (such as range, variance, and standard deviation), and creating visual representations (such as histograms, box plots, and scatter plots).

How do you determine if a distribution is normal?

A distribution is considered normal if it follows a bell-shaped curve and has certain characteristics, such as a symmetric shape, a single peak, and the mean, median, and mode all being equal. This can be visually assessed through a histogram or mathematically tested through methods such as the Kolmogorov-Smirnov test or the Shapiro-Wilk test.

Can the distribution of n random variables change over time?

Yes, the distribution of n random variables can change over time. This can be due to various factors such as changes in the underlying data, changes in the population being studied, or external factors influencing the data. It is important to regularly monitor and update the description of the distribution to reflect any changes that may occur.

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