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**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.

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