The definition of random sample mean and convergence

[bgd2007]

Hi.
I have the following question:
I am a student and my teacher defined the random sample as following:

Def:A simple and repeated random sample of size n from the population (omega,K)
for the characteristic Xmega->R is a random vector
V=(X_1,X_2,...,X_n) defined on (omega)^n and with values in R^n as follows:

for O_n=(o_1,o_2,...,o_n) in omega^n we define X_i(O_n)=X(o_i)
Then he proves that X_1,...,X_n are stochastic independent and have the same repartition as the characteristic for study,X,and he calls them sample random variables
Things are well till now.
Then he defines a new "sample random variable", called "the sample mean" like this:

m(X,O_n)mega^n ->R

m(X,O_n)=[X_1(O_n)+X_2(O_n)+...+X_n(O_n)]/n.
Good.

Now(and here is the problem) he states that the
sequence of random variables m(X,O_n) for n converges almost sure to M(X) the mean of the characteristic.
What i don't understand is the follwing:
how is this a sequence of random variables?
I mean,when we define a sequence of functions don't they have the same domain?
Here, m(X,O_n) has the domain omega^n, which changes with n!
Isn't this a problem??
I looked on the definition of almost sure convergence and it says that all X_n are defined on the same set!

And another problem is the proof of this teacher for the almost sure convergence of the sequence m(X;O_n)-
he applies the law of large numbers to the sequence X_n!
But how the hell can he do that,since
in

m(X;O_n) we have X_1,...,X_n defined on omega^n and in
m(X;O_n+1) the random variables X_1,...X_n,X_n+1 are defined on omega^(n+1)
so X_1,...,X_n in m(X;O_n+1) are different from X_1,...,X_n in omega^n.
Is this teacher right?

 

[EnumaElish]

Good points; why don't you raise them with your teacher?

 

[tacman]

Hi there,

I understand your confusion and it is a common one when first learning about random samples and convergence. Let me break it down for you.

A random sample is a selection of individuals from a larger population that is chosen randomly. This means that each individual in the population has an equal chance of being selected for the sample. This sample is then used to make inferences about the larger population.

The sample mean is a statistic calculated from the random sample, and it represents the average value of the characteristic being studied in the sample. In your example, the sample mean is denoted as m(X,O_n). This is a random variable because it is calculated from the random sample, and the values of the sample mean will vary depending on the individuals selected for the sample.

Now, when we talk about convergence, we are talking about the behavior of a sequence of random variables. In this case, the sequence is m(X,O_n) for n. This means that as n increases (i.e. as we take larger and larger random samples), the values of m(X,O_n) will converge to a certain value. In this case, that value is the mean of the characteristic being studied, denoted as M(X).

It is true that the domain of m(X,O_n) changes with n, but this is not a problem. As long as the domain of each m(X,O_n) is the same, which it is (omega^n), then we can talk about convergence. It is also important to note that the values of m(X,O_n) do not change with n, only the size of the sample changes.

As for the proof of almost sure convergence, your teacher is correct in using the law of large numbers. This law states that as n approaches infinity, the sample mean will converge to the population mean. In this case, as n approaches infinity, the sequence m(X,O_n) will converge to M(X), the mean of the characteristic being studied in the population.

I hope this helps to clarify the concept of random sample mean and convergence for you. Remember, a random sample is a selection of individuals from a larger population, and the sample mean is a random variable calculated from that sample. Convergence refers to the behavior of a sequence of random variables, in this case, the sample mean.