## Summing a Distribution to Find a Constant

The one thing that all probability distributions have in common is that all probabilities add to 1.
Suppose a random variable
$X$
follows the distribution
$P(X=x)=a (\frac{2}{5})^x$
, where
$a$
is an unknown constant. We can find
$a$
using the condition
$\sum_x P(X=x)=1$
.
$\sum_{x=0}^{\infty} a(\frac{2}{5})^x=1$
.
The sequence
$P(X=0)=a, \: P(X=1)=\frac{2}{5}a, ..., P(X=k)=(\frac{2}{5})^k a$
is a geometric sequence with first term
$a$
and common ration
$\frac{2}{5}$
so we can use the formula for the sum of a geometric sequence
$S=\frac{a}{1-r}$
.
$1=\frac{a}{1-2/5} =\frac{a}{3/5} \rightarrow a= \frac{3}{5}$
.