Why does Binomial dist. converge in distribution to Poisson dist. ?

  1. Hey guys,

    In class, I was shown that the Binomial prob density function converges to the Poisson prob density function. But why does this show that the Binomial distribution converges in distribution to the Poisson dist. ? Convergence in distribution requires that the cumulative density functions converges (not necessarily the prob density functions).

    Is it because both are non negative and start at zero --> therefore convergence in prob. density functions implies converges in cumulative density functions?

    Thank you.
     
    Last edited: Aug 18, 2012
  2. jcsd
  3. Hey jojay99 and welcome to the forums.

    For this example, you should let the number of events in an interval go to infinity but keep the independence property of the binomial that each event is independent and this translates into the property that they do not overlap.

    So in the binomial we considered getting so many successes which can be thought of as the number of times an event happens within an interval. But each interval is completely disjoint from every other.

    So we take a limit where we consider the distribution of an infinite number of events being possible in that particular interval. In the binomial we had n+1 possibilities ranging from 0 to n, but now we are letting that become infinitely many in one single non-overlapping interval.

    The identity to get the exponential relates to the limit form of (1 + 1/n)^n.

    I did a google search that does it much better than me, but hopefully the above gives you a non-mathematical explanation of what is going on.

    http://www.the-idea-shop.com/article/216/deriving-the-poisson-distribution-from-the-binomial
     
  4. So convergence in the probability density functions implies that the cdfs converges?
     
  5. The CDF is uniquely determined by the PDF so there is a one to one correspondence with a PDF to the CDF. We also assume the PDF is a proper PDF satisfying the Kolmogorov Axioms.
     
  6. Stephen Tashi

    Stephen Tashi 4,417
    Science Advisor
    2014 Award

    There are several types of convergence defined for sequences of functions and there are several types of convergence defined for random variables. So you'd have to say what type of convergence you're talking about before I'd believe that implication.

    I think this PDF deals with you question: http://www.google.com/url?sa=t&rct=...sg=AFQjCNGYITECDXIxnnXU7geHq5-VIJQQLw&cad=rja On page 2, it cites "Scheffe's Theorem" as proving that "pointwise" convergence of a sequence of discrete PDFs to a discrete PDF implies convergence "in distribution" of the associated CDFs.

    You are correct that what was shown in your class was not a sufficient proof. And, of course, Scheffe's theorem itself requires a proof.
     
    Last edited: Aug 18, 2012
  7. Thank you guys for your help.

    Yeah, I was referring to point wise convergence of the pdfs.

    I always thought that Scheffe's Theorem only applied to continuous random variables. I guess I'm wrong.
     
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