Truncated harmonic series?

Probability Level 2

Let $Y$ be a discrete random variable having the following probability density function,

$P(Y =y) = \dfrac1{y} ,$

where $y = n, n+1, n + 2 ,\ldots$ and $n$ is a positive integers . What is the number of digits of $n$ ?

This is an impossible scenario 4 3 5

1 solution

Gonçalo Vieira
Oct 24, 2019

An important property of discrete random variables is that the sum of the probabilities of its possible values is equal to 1. So:

$\frac{1}{n}$ + $\frac{1}{n+1}$ + $\frac{1}{n+2}$ + ... = 1

but the left side is the harmonic series without the first n-1 terms. Noticing that:

$\sum_{k=1}^{\infty} a(k)$ converges if and only if $\sum_{k=n}^{\infty} a(k)$ converges for any $n\in\N$

As the harmonic series diverges, for any $n\in\N$ the right side will diverge, so it is impossible.