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

In summary, convergence in probability density functions implies convergence in cumulative density functions.
  • #1
jojay99
10
0
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.
 
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  • #2
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
 
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  • #3
So convergence in the probability density functions implies that the cdfs converges?
 
  • #4
jojay99 said:
So convergence in the probability density functions implies that the cdfs converges?

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.
 
  • #5
jojay99 said:
So convergence in the probability density functions implies that the cdfs converges?

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.
 
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  • #6
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.
 

1. What is the Binomial distribution and the Poisson distribution?

The Binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials with a constant probability of success. The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space.

2. Why does the Binomial distribution converge in distribution to the Poisson distribution?

The Binomial distribution can be thought of as a special case of the Poisson distribution, where the number of trials approaches infinity and the probability of success approaches 0 in such a way that the mean remains constant. This convergence is known as the law of rare events.

3. What is the significance of the convergence of the Binomial distribution to the Poisson distribution?

This convergence allows for easier calculation and approximation of probabilities in situations where the number of trials is large and the probability of success is small. It also allows for the use of Poisson distribution in situations where the assumptions of the Binomial distribution are not met.

4. Are there any limitations to the convergence of Binomial distribution to Poisson distribution?

The convergence is only valid when the number of trials is large and the probability of success is small. If these conditions are not met, the convergence may not hold and the use of Poisson distribution may not be appropriate.

5. Can the convergence of Binomial distribution to Poisson distribution be proven?

Yes, the convergence can be proven mathematically using the Central Limit Theorem. This theorem states that as the number of trials increases, the Binomial distribution approaches a normal distribution, and as the probability of success decreases, the normal distribution approaches a Poisson distribution.

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