Confused with basic terms:iid, rv

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September 16th 2008, 07:38 PM
chopet

Confused with basic terms:iid, rv

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

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.
$x_1, x_2, x_3...x_n$ are not random variables, but values which the random variable (a function) assumes.

So here's my confusion.

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

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

Thanks for your time!
September 17th 2008, 04:14 AM
Laurent

Quote:

Originally Posted by chopet

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

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, $x_1,\ldots,x_n$ are independent copies of a random variable $X$ , i.e. they are independent identically distributed random variables (and their common distribution is that of $X$ ). This is the way to model " $n$ values drawn from a random variable $X$ ", as you wrote. For a given sample $\omega$ , a random variable assumes only one value $X(\omega)$ . 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 $x_1(\omega),\ldots,x_n(\omega)$ for some $\omega$ in the sample space.

I hope this makes things clearer...
Laurent.
September 17th 2008, 04:40 AM
chopet

yes, they do. Thanks.