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Math Help - Confused with basic terms:iid, rv

  1. #1
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    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!
    Last edited by chopet; September 16th 2008 at 07:52 PM.
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  2. #2
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    Quote Originally Posted by chopet View Post
    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.
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  3. #3
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    yes, they do. Thanks.
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