Proving the Limit of Binomial Distribution as Poisson: Formal Proof

  • Thread starter relinquished™
  • Start date
  • Tags
    Limit
In summary, the speaker is stuck in their proof of showing the limit of a binomial distribution is the Poisson distribution. They are questioning how the expression approaches 1 as n gets larger and are looking for a formal proof. Their approach is to group terms together in the numerator and denominator.
  • #1
relinquished™
79
0
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:

[tex]

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

[/tex]

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
 
Physics news on Phys.org
  • #2
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.
 
  • #3



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.
 

1. What is the binomial distribution and its relationship to the Poisson distribution?

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials. It is related to the Poisson distribution because as the number of trials increases, the binomial distribution approaches the Poisson distribution. In other words, the Poisson distribution can be thought of as a limiting case of the binomial distribution.

2. How can the limit of the binomial distribution be proven to be the Poisson distribution?

The limit of the binomial distribution can be proven to be the Poisson distribution through the use of mathematical techniques such as Taylor series expansions and the Central Limit Theorem. These methods allow us to demonstrate that as the number of trials increases, the binomial distribution becomes increasingly similar to the Poisson distribution.

3. What assumptions are necessary for the limit of the binomial distribution to be the Poisson distribution?

The assumptions necessary for the limit of the binomial distribution to be the Poisson distribution include having a fixed number of independent trials, a constant probability of success for each trial, and a finite number of trials. Additionally, the number of trials must be sufficiently large in order for the limit to hold.

4. Can the limit of the binomial distribution be proven for all values of the parameters?

Yes, the limit of the binomial distribution can be proven for all values of the parameters as long as the assumptions mentioned above are met. However, as the parameters approach certain extreme values, the limit may not hold due to the discontinuity of the Poisson distribution at those points.

5. How is the limit of the binomial distribution used in real-world applications?

The limit of the binomial distribution is used in various real-world applications such as quality control, where it is used to model the number of defective items in a batch. It is also used in genetics to model the number of mutations in a given DNA sequence. Additionally, the Poisson distribution is used in insurance and risk management to model rare events such as accidents or natural disasters.

Similar threads

  • Calculus and Beyond Homework Help
Replies
13
Views
637
  • Calculus and Beyond Homework Help
Replies
6
Views
421
  • Calculus and Beyond Homework Help
Replies
8
Views
588
  • Calculus and Beyond Homework Help
Replies
17
Views
486
  • Math POTW for University Students
Replies
3
Views
558
  • Calculus and Beyond Homework Help
Replies
2
Views
658
Replies
14
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
Replies
4
Views
1K
  • Calculus
Replies
4
Views
1K
Back
Top