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Math Help - Law of large numbers, an exercise

  1. #1
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    Law of large numbers, an exercise

    This isn't advanced statistics but I think this subforum is better to post this question in:

    I have a lab that's supposed to illustrate the law of large numbers (LLN), and we're using Matlab (this is not a Matlab question). First step is to create a "set of data with 300 observations from an exponential distribution with expected value 1", chanceno = exprnd(1,1,300). Then we create "a vector 'avg' with 300 elements where i is the average of the i first random numbers and produces a plot of this vector";

    sums = cumsum(chanceno)
    for i=1:length(chanceno)
    avg(i)=sums(i)/i;
    end
    plot(avg)

    This does indeed produce such a plot, which seems to converge to 1, the expected value, as per the law of large numbers. Now to my question: How does this illustrate LLN? My interpretation of all this (which was wrong) was that we're creating a dataset of 300, then one of these 300 is chosen according to an exponential distribution and repeated a bunch of times. I now realize this is wrong, very wrong, but I'm not sure why this exercise is about LLN. My problem is with "random numbers"; the formulation of LLN which I have in my book is that we have sequence of independent random (or stochastic) variables, etc etc. Should I view the generation of each of these 300 random numbers as a random variable? I.e. in the formulation of LLN I should view each of the 300 random numbers as an X_i, as in; "let X1,X2,... be a sequence of independent random variables with the same expected value mu and std.dev. sigma. Let \overline{X}_n := (1/n) \sum_{i=1}^n X_i be the average of the n first variables. Then for all \epsilon > 0, Pr(|\overline{X}_n - mu| > \epsilon) -> 0 as n -> \infty"

    This is all a bit vague I suppose, but I hope someone understands what I'm asking. Thanks in advance.
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  2. #2
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    Re: Law of large numbers, an exercise

    Hey spudwish.

    Every random number should be treated as a random variable drawn from a particular distribution (in this case, exponential).

    This is actually the basic idea behind all of statistics: every element in the sample has a particular distribution and from that we derive a set of statistics (which is just a function of random variables).
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  3. #3
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    Re: Law of large numbers, an exercise

    Alright, then I'm on the right track at least. Thank you!
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