Proving the Limit of Binomial Distribution as Poisson: Formal Proof

[relinquished™]

I'm a little stuck in my proof here. As I was trying to prove that the limit of a binomial distribution is the poisson distribution, I encountered this:

<br /> <br /> \lim_{n\to +\infty} \frac{n!}{(n-x)! (n-k)^x}<br /> <br />

where x and k are arbitrary constants.


The books say that this approaches 1, but shows no formal proof. How are we sure that this approaches 1 as n gets larger? In short, what's the formal proof?

Thanx for any help

 

[Hurkyl]

Let's see... you have n things multiplied together on the top (1, 2, 3, ..., n), and you have n things multipled together on the bottom: (n - x) of them in (n-x)! and x of them in (n-k)^x. My first instinct would be to try and group terms in the numerator with terms in the denominator.

 

[M98Ranger]

To prove that the limit of the binomial distribution is the Poisson distribution, we need to show that the expression \frac{n!}{(n-x)! (n-k)^x} approaches 1 as n approaches infinity. This can be done by using the definition of a limit.

First, let's rewrite the expression as \frac{n(n-1)(n-2)...(n-x+1)}{(n-k)^x}. We can then rewrite this as \frac{n^x}{(n-k)^x} \cdot \frac{(n-1)(n-2)...(n-x+1)}{n(n-1)(n-2)...(n-x+1)}. Notice that the first fraction approaches 1 as n approaches infinity, since the numerator and denominator both have a factor of n that cancels out. So, we can focus on the second fraction.

Using the property of limits, we can rewrite the second fraction as \frac{(n-1)x}{n(n-1)(n-2)...(n-x+1)} \cdot \frac{(n-2)...(n-x+1)}{(n-1)x}. Now, as n approaches infinity, we can see that the first fraction approaches 0, since the numerator is a constant (n-1)x and the denominator is increasing without bound. The second fraction also approaches 1, since the numerator and denominator both have a factor of n that cancels out.

Therefore, the entire expression approaches 1 as n approaches infinity. This proves that the limit of the binomial distribution is the Poisson distribution.

I hope this helps clarify the formal proof for you. If you have any further questions, don't hesitate to ask.