Proving Distribution Equality Using Stirling Approximation

[physics.alex]

Homework Statement

I am asked to prove the following distribution equal to 1. The distribution is obtained by using classic Binomial distribution and apply stirling approximation.

Homework Equations

P(X) = \frac{1}{\sqrt{Nx(1-x)}}exp(-NI(x))

where I(x) = x ln (x/p) + (1-x) ln (1-x/1-p)


This form seems complicated to me. Any suggest for the first step to simplify the proof??

 

[Mark44]

I would say substitute the expression you have for I(x) into the expression you have for P(X), but what you have for I(x) is ambiguous, especially this part:
ln(1 - x/1 - p)

If I take this at face value, it is
ln(1 - \frac{x}{1} - p)

I don't think that's what you meant, though, so I will have to interpret what you have written.
Is it
ln(\frac{1 - x}{1 - p})?

Or is it
ln(1 - \frac{x}{1 - p})?

 

[physics.alex]

sorry for unclear formula.
it should be

ln{ (1-x) / (1-p) }

*I am not familiar with the latex format sorry..
Alex

I would say substitute the expression you have for I(x) into the expression you have for P(X), but what you have for I(x) is ambiguous, especially this part:
ln(1 - x/1 - p)

If I take this at face value, it is
ln(1 - \frac{x}{1} - p)

I don't think that's what you meant, though, so I will have to interpret what you have written.
Is it
ln(\frac{1 - x}{1 - p})?

Or is it
ln(1 - \frac{x}{1 - p})?

 

[Mark44]

Whether you know LaTeX or not you should be aware of how to write rational expressions so that they are not ambiguous; that is, by using enough parentheses in the right places so that their meaning is clear.

For your problem, work with I(x) using the properties of logs. From this, you should get the log of the product of two expessions involving the fractions x/p and (1 - x)/(1 - p).

From there, replace the two factors by their Stirling approximations.

I haven't worked this all the way through, but that's the direction I would take.

 

[physics.alex]

Thanks and will try it.
Alex