# Stirlings approx/CoinFlips/Gamma function

• thatguy14
In summary, the conversation discusses three questions: proving a formula for ln(n!), calculating the probability of getting exactly 500 heads when flipping 1000 coins, and showing an integral formula for factorial. The person asking for help is lost on the first and third questions and needs clarification on the second.
thatguy14

## Homework Statement

Hi I actually have three questions that I am posting here, help in all of them would be greatly appreciated!

1) Prove that ln(n!) ≈ nln(n)-n+ln(2*pi*n)/2 for large n

2)Supposed you flip 1000 coins, what is the probability of getting exactly 500 heads

3) Show that n! = $\int x^{n}exp(-x)dx$ where n is an integrer and the injtegral is from 0 to infinity. (The gamma function extends this definition of factorial to include non-integrers, z).

Below

## The Attempt at a Solution

1) I am pretty lost here. This is a third year thermodynamics course (not been heavily math based so far) so the solution shouldn't be something above that level. I don't even know where to start.

2) This one I thought would be really easy (and it should be). This is what I did.

I started off by ignoring the last term of stirlings approximation (this is what the textbook has and my prof did). the probability is

P$_{N}$(n) = $\frac{N!}{(N-n)!n!}$*2$^{-N}$
by using stirlings approximation, (first taking the natural log of both sides) and simplifying I got (and my professor)

P = exp(NlnN - (N-n)ln(N-n)-nln(n)-Nln(2))

but when I plug in 1000 for N and 500 for n I keep getting 1 which I don't think is right. What am I doing wrong?

3)I tried to do an integration by parts on the right side but it leads to some undefined things (i.e. 0*infinity) so I am lost yet again. I thought that was just a definition, how do I show that?

Thanks

Please ignore number two, it required that the terms left off to be added. Still need help with 1 and 3

## 1. What is Stirling's approximation?

Stirling's approximation is a mathematical formula used to approximate the factorial of a large number. It is expressed as n! ≈ √(2πn)(n/e)^n, where n is the large number and e is the base of the natural logarithm. This formula provides a good estimate for n! when n is very large.

## 2. How is Stirling's approximation used in science?

Stirling's approximation is commonly used in many scientific fields, particularly in statistics and physics, to simplify complex calculations involving factorials of large numbers. It is also used in the analysis of algorithms and in the study of asymptotic behavior of functions.

## 3. What are some limitations of Stirling's approximation?

Stirling's approximation is only accurate for large values of n. As n gets smaller, the approximation becomes less accurate. Additionally, Stirling's approximation does not account for the oscillating behavior of factorials for certain values of n, resulting in some inaccuracies in its estimation.

## 4. What is the connection between coin flips and Stirling's approximation?

In probability theory, Stirling's approximation is often used to calculate the number of possible outcomes for a large number of coin flips. For example, if you flip a coin 100 times, Stirling's approximation can be used to estimate the number of possible outcomes (2^100).

## 5. What is the gamma function and how is it related to Stirling's approximation?

The gamma function is a mathematical function that extends the concept of factorial to real and complex numbers. Stirling's approximation is often used to approximate the gamma function for large values of its argument. This relationship is commonly used in mathematical and scientific calculations involving the gamma function.

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