# Thread: a binomial distribution proof

1. ## a binomial distribution proof

Show that if $\displaystyle X$~$\displaystyle B(n,p)$, then

$\displaystyle P(X=x+1)=((n-x)/(x+1))*(p/(1-p))*P(X=x),$ and $\displaystyle x=0,1,2, ..., n-1$

2. Originally Posted by shawli
Show that if $\displaystyle X$~$\displaystyle B(n,p)$, then $\displaystyle P(X=x+1)=((n-x)/(x+1))*(p/(1-p))*P(X=x),$ and $\displaystyle x=0,1,2, ..., n-1$
Recall that $\displaystyle P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
Now observe that $\displaystyle \left(\frac{p}{1-p}\right) \left(\frac{n-x}{x+1}\right)P(X=x)=\left(\frac{p}{1-p}\right) \left(\frac{n-x}{x+1}\right)\binom{n}{x}p^x(1-p)^{n-x}$
Now combine and reduce to get to the LHS.