a binomial distribution proof

**shawli** · April 2nd 2010, 01:04 PM

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

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

**Plato** · April 2nd 2010, 02:33 PM

Originally Posted by shawli

Show that if $X$ ~ $B(n,p)$ , then $P(X=x+1)=((n-x)/(x+1))*(p/(1-p))*P(X=x),$ and $x=0,1,2, ..., n-1$

Recall that $P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
Now observe that $\left(\frac{p}{1-p}\right) \left(\frac{n-x}{x+1}\right)P(X=x)=\left(\frac{p}{1-p}\right) \left(\frac{n-x}{x+1}\right)\binom{n}{x}p^x(1-p)^{n-x}$
Now combine and reduce to get to the LHS.

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