## Summing a Distribution to Find a Constant

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 \[S=\frac{a}{1-r}\]

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

.