Confused with basic terms:iid, rv

• Sep 16th 2008, 07:38 PM
chopet
Confused with basic terms:iid, rv
$\displaystyle x_1, x_2, x_3...x_n$ are iid rvs with $\displaystyle E[x_i] = \mu$ and $\displaystyle Var[x_i] = \sigma^2$

I need to make sure my understanding of this statement is correct.
If X is a rv, then it is a function defined over a sample space.
i = 1,2,3,....n constitute the number of trials of this rv.
$\displaystyle x_1, x_2, x_3...x_n$ are not random variables, but values which the random variable (a function) assumes.

So here's my confusion.

If $\displaystyle x_1$ is only a value, analogous to the range of a function, say, $\displaystyle f(x) = x^2$, what is the actual meaning of
$\displaystyle E[x_1] = \mu$ and $\displaystyle Var[x_1] = \sigma^2$? Is it just trying to imply that these values are all drawn from a RV whose parameters are $\displaystyle E[X] = \mu$ and $\displaystyle Var[X] = \sigma^2$?

Are they liberal with the semantics or is my understanding wrong?

• Sep 17th 2008, 04:14 AM
Laurent
Quote:

Originally Posted by chopet
$\displaystyle x_1, x_2, x_3...x_n$ are iid rvs with $\displaystyle E[x_i] = \mu$ and $\displaystyle Var[x_i] = \sigma^2$

I need to make sure my understanding of this statement is correct.

Usually, random variables are written in block capitals, that's why this definition is a bit confusing. In fact, $\displaystyle x_1,\ldots,x_n$ are independent copies of a random variable $\displaystyle X$, i.e. they are independent identically distributed random variables (and their common distribution is that of $\displaystyle X$). This is the way to model "$\displaystyle n$ values drawn from a random variable $\displaystyle X$", as you wrote. For a given sample $\displaystyle \omega$, a random variable assumes only one value $\displaystyle X(\omega)$. When modeling several independent repetitions an experiment (like throwing a coin), the sample space has to be widened so that it contains the values of every sequence that can appear. Numerical observations would be $\displaystyle x_1(\omega),\ldots,x_n(\omega)$ for some $\displaystyle \omega$ in the sample space.

I hope this makes things clearer...
Laurent.
• Sep 17th 2008, 04:40 AM
chopet
yes, they do. Thanks.