# Maximal number of independent random variables

• fishy_1980
In summary, the conversation discusses the maximum number of independent binary random variables in a sample space with 2^n points. It is suggested to use mathematical induction to prove that the maximum number is n, but there are doubts about this statement. It is also mentioned that dependent random variables can be defined algebraically or in terms of linear dependence. Ultimately, it is concluded that the maximum number of independent binary random variables is 2^n and not n.
fishy_1980
Hi all,

assume we have a sample space with 2^n points. (it size is 2^n for some natural n)
I need to prove that the maximal number of independent binary (indicator) random variables (which are not trivial, i.e. constant) is n...

Thnks,
Pitter

Have you tried mathematical inuction on n. It seems pretty straightforward.

I am not sure that I understand.

Say I have 4 obs with 3 independent variables:

y x1 x2 x3
10 1 0 1
11 1 0 0
12 0 1 0
20 0 1 1

I can estimate the model y = b1 x1+ b2 x2 + b3 x3 + u with the following results:
b1 = 8.75
b2 = 14.25
b3 = 3.5

I've tried induction,

but I still don't get it...

I don't really know how to algebraically define
dependent random variables (dependent by their random variable function)

I am not convinced that the statement is true. Did you see my example?

n = 3
2^n = 8

Since I can uniquely and independently estimate a 3-variable model with 4 observations, I can surely estimate, say, a 4-variable model with 8 observations. Yet the statement implies that with 8 observations the maximal number of variables is 3.

Another example with 2^3 = 8 observations and 4 independent binary variables:
x1 x2 x3 x4
1 0 1 0
1 0 0 1
0 1 0 1
0 1 1 1
1 0 1 1
1 0 0 1
0 1 0 1
0 1 1 0

Last edited:
fishy_1980 said:
I don't really know how to algebraically define
dependent random variables (dependent by their random variable function)
Random variables X and Y are dependent if Prob{X = x and Y = y} $\ne$ Prob{X=x}Prob{Y=y}, or equivalently, Prob{X < x and Y < y} $\ne$ Prob{X<x}Prob{Y<y}.

Alternatively, variables x1, x2, x3 and x4 are linearly dependent if there exists a1, a2, a3 such that x4 = a1 x1 + a2 x2 + a3 x3. Equivalently, they are lin. dep. if there exists a1, a2, a3 and a4, not all equal to zero, such that a1 x1 + a2 x2 + a3 x3 + a4 x4 = 0.

Neither of my examples above is a case of linear dependence.

Last edited:
thanks EnumaElish,

I think you're right,
and if looking at it that way (as vector space) the maximum number of independent binary random variables above 2^n points in the sample space, is 2^n...and not n.

## What is the definition of maximal number of independent random variables?

The maximal number of independent random variables refers to the largest number of random variables that can be simultaneously and independently observed in a given scenario.

## Why is the maximal number of independent random variables important?

Understanding the maximal number of independent random variables is crucial in various fields of science, such as statistics, probability, and data analysis. It allows researchers to determine the maximum amount of information that can be obtained from a set of variables and make accurate predictions based on this information.

## How is the maximal number of independent random variables calculated?

The calculation of the maximal number of independent random variables depends on the dimensionality of the problem. For example, in a two-dimensional scenario, the maximum number of independent random variables is two, while in a three-dimensional scenario, it is three. In general, the maximal number of independent random variables is equal to the number of dimensions of the problem.

## What factors can affect the maximal number of independent random variables?

The maximal number of independent random variables can be affected by various factors, such as the complexity of the problem, the amount of available data, and the level of correlation between the variables. In some cases, a smaller number of independent variables may be sufficient to represent the data accurately.

## Can the maximal number of independent random variables change over time?

Yes, the maximal number of independent random variables can change over time, especially in dynamic systems where new variables may emerge or existing ones may become dependent on each other. It is important to regularly reassess the maximal number of independent random variables in such cases to ensure accurate data analysis and predictions.

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