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 \[S=\frac{a}{1-r}\]
.\[1=\frac{a}{1-2/5} =\frac{a}{3/5} \rightarrow a= \frac{3}{5}\]
.