Binomial Distribution satisfies Marcoff Chain

[ppedro]

1. The problem statement

Consider the Binomial Distribution in the form

[itex]P_{N}(m)=\frac{N!}{(\frac{N+m}{2})!(\frac{N-m}{2})!}p^{\frac{N+m}{2}}q^{\frac{N-m}{2}}[/itex]

where [itex]p+q=1[/itex], [itex]m[/itex] is the independent variable and [itex]N[/itex] is a parameter.

Show that it satisfies the marcoff chain

[itex]P_{N+1}\left(m\right)=pP_{N}\left(m-1\right)+qP_{N}\left(m+1\right)[/itex]

2. The attempt at a solution

I'm trying my solution starting from this:

[itex]pP_{N}\left(m-1\right)+qP_{N}\left(m+1\right)[/itex]


[itex]=p\frac{N!}{\left(\frac{N+m-1}{2}\right)!\left(\frac{N-m+1}{2}\right)!}p^{\frac{N+m-1}{2}}q^{\frac{N-m+1}{2}}+q\frac{N!}{\left(\frac{N+m+1}{2}\right)!\left(\frac{N-m-1}{2}\right)!}p^{\frac{N+m+1}{2}}q^{\frac{N-m-1}{2}}[/itex]


[itex]=\frac{N!}{\left(\frac{N+m-1}{2}\right)!\left(\frac{N-m+1}{2}\right)!}p^{\frac{N+m+1}{2}}q^{\frac{N-m+1}{2}}+\frac{N!}{\left(\frac{N+m+1}{2}\right)!\left(\frac{N-m-1}{2}\right)!}p^{\frac{N+m+1}{2}}q^{\frac{N-m+1}{2}}[/itex]


[itex]=p^{\frac{N+m+1}{2}}q^{\frac{N-m+1}{2}}\left(\frac{N!}{\left(\frac{N+m-1}{2}\right)!\left(\frac{N-m+1}{2}\right)!}+\frac{N!}{\left(\frac{N+m+1}{2}\right)!\left(\frac{N-m-1}{2}\right)!}\right)[/itex]


I can't go any further. If you can help I would appreciate.

 

[TSny]

(Edit) See what happens if you multiply by##\frac{N+1}{N+1}##.

 

[TSny]

My previous comment was based on a particular way that I went about it which got to the result. But, on review, I see that multiplying by (N+1)/(N+1) isn't necessary.

I would factor out the ##N!## in your expression.

The important thing is to get the two denominators in your expression to match the denominator in ##P_{N+1}(m)##. For example, what could you multiply the numerator and denominator of ##\frac{1}{\left(\frac{N+m-1}{2}\right)!\left(\frac{N-m+1}{2}\right)!}## by to get the denominator in ##P_{N+1}(m)##?

[By the way, welcome to PF, ppedro!]

 

[ppedro]

Hey TSny! Thanks for your reply. I see what you're suggesting but I'm not being able to compute it. The factorials are not helping me simplify the expression.

 

[TSny]

##P_{N+1}(m)## has a denominator of ##(\frac{N+m+1}{2})!(\frac{N-m+1}{2})!##.

Compare that to your first denominator ##(\frac{N+m-1}{2})!(\frac{N-m+1}{2})!##.

To get your denominator to match the denominator in ##P_{N+1}(m)##, you've got to somehow transform ##(\frac{N+m-1}{2})!## into ##(\frac{N+m+1}{2})!##.

What can you multiply ##(\frac{N+m-1}{2})!## by to produce ##(\frac{N+m+1}{2})!##?

 

[ppedro]

[itex]\frac{(N+m+1)!}{(N+m-1)!}=\frac{(N+m+1)(N+m)(N+m-1)!}{(N+m-1)!}=(N+m+1)(N+m)
[/itex]

 

[TSny]

No. Note that ##\frac{N+m+1}{2} = \frac{N+m-1}{2} + 1##.

 

[ppedro]

Ok, I see your point. Thanks!