Question

Let x be a random variable with pmf pk = c/k2 for k = 1, 2, …a. estimate the value of c numerically. note that the series converges

Answer 1

$f_X(x)=\begin{cases}\dfrac c{k^2}&\text{for }k\in\{1,2,\ldots\}\\0&\text{otherwise}\end{cases}$

Because this is a valid PMF, you must have

$\displaystyle\sum_{k\ge1}f_X(x)=\sum_{k\ge1}\frac c{k^2}=1$

Recall that

$\displaystyle\sum_{k\ge1}\frac1{k^2}=\dfrac{\pi^2}6$

so the above sum is

$\displaystyle\sum_{k\ge1}\frac c{k^2}=\frac{c\pi^2}6=1$

which means that you must have

$c=\dfrac6{\pi^2}\approx0.6079$