Simplifying Summation and Factorial

In summary, the Poisson distribution can be derived as the limiting case of the binomial distribution using simplification steps such as splitting the last factor, moving 1/k! into the second to last factor, and writing n!/(n-k)! explicitly as k terms while taking the (1/n)^k from the second to last term. The main simplification comes from fixing k and
  • #1
rwinston
36
0
I was looking at the web page containing a derivation for the Poisson distribution:

http://en.wikipedia.org/wiki/Poisson_distribution

which derives it as the limiting case of the binomial distribution. There is a simplification step which I am missing, which is the step(s) between

[tex]
\lim_{n\rightarrow\infty}\frac{n!}{k!(n-k)!}\left(\frac{\lambda}{n}\right)^k\left(1-\frac{\lambda}{n}\right)^{n-k}
[/tex]

and

[tex]
=\lim_{n\to\infty} \underbrace{\left({n \over n}\right)\left({n-1 \over n}\right)\left({n-2 \over n}\right) \cdots \left({n-k+1 \over n}\right)}\ \underbrace{\left({\lambda^k \over k!}\right)}\ \underbrace{\left(1-{\lambda \over n}\right)^n}\ \underbrace{\left(1-{\lambda \over n}\right)^{-k}}
[/tex]

Does the main simplification come from:

[tex]
\frac{n!}{k!(n-k)!} \left( \frac{\lambda}{n} \right)^k
[/tex]

[tex]
=\frac{(n-k+1)!\lambda^k}{k!n^k}
[/tex]

?
 
Last edited:
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  • #2
No, too complicated. The "simplifications" they have done are
1) split the last factor in two with powers n and -k
2) move 1/k! into the 2nd to last factor
3) write n!/(n-k)! explicitly as k terms while stealing the (1/n)^k from the 2nd to last term.
 
  • #3
If k is fixed, (n/n)...((n-k+1)/n) ->1, (1-L/n)n) -> e-L, and (1-L/n)-k ->1.

This results in the Poisson term for k.
 
  • #4
[tex]\frac{n!}{k!(n-k)!}\left(\frac{\lambda}{n}\right)^n= \frac{\lambda^k}{k!}\frac{n!}{(n-k)!n^k}= \frac{\lambda^k}{k!}\frac{n(n-1)\cdot\cdot\cdot(n-k+1)}{n^k}= \frac{\lambda^n}{k!}\frac{n}{n}\frac{n-1}{n}\cdot\cdot\cdot\frac{n-k+1}{n}[/tex]
 

1. What is the purpose of simplifying summation and factorial in mathematics?

Simplifying summation and factorial helps us to easily calculate and manipulate large numbers and equations. It also helps to identify patterns and relationships between numbers.

2. What is summation and how is it simplified?

Summation, denoted by the symbol ∑, is a mathematical operation that represents the addition of a sequence of numbers. It is simplified by using formulas, properties, and rules to reduce the number of terms in the sequence.

3. What is factorial and how is it simplified?

Factorial, denoted by the symbol !, is a mathematical operation that represents the product of a sequence of consecutive numbers. It is simplified by applying the formula n! = n * (n-1) * (n-2) * ... * 2 * 1, where n is a positive integer.

4. Why is it important to simplify summation and factorial?

Simplifying summation and factorial allows us to solve complex mathematical problems efficiently, especially in areas such as probability, statistics, and combinatorics. It also helps to understand and prove mathematical concepts and theories.

5. Can summation and factorial be simplified in all cases?

No, there are certain cases where it is not possible to simplify summation and factorial. For example, when dealing with infinite sequences or when the numbers in the sequence are not consecutive integers.

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