Concluding that a sum is monotone increasing in an argument

Question

Let $f_i (x)$, $i \geq 1$, $x \in \mathbb{R}$, be a sequence of positive functions. Suppose that $$ \frac{f_{i+1} (x)}{f_i(x)} $$ increasing in $x$ is. Moreover, suppose $$ \sum_{i=1}^{\infty} f_i(x) = 1.$$ I want to conclude that then $$ \sum_{i=1}^\infty i f_i(x)$$ also montonely increasing in $x$ is. It somehow makes sense but I can not argue rigorously. Any ideas?

Answer 1

Suppose you have a random variable $X$, and let $f_i(x)$ denote the probability that the event $\{X = i\}$ occurs. These events are disjoint. Note that we have the normalization condition $\sum_i f_i(x) = 1$, which permits this interpretation.

The increasing ratio condition means that, as $x$ increases, the probabilities are further and further spaced towards larger values (i.e. $f_5(x)$ becomes a lot more probable than $f_4(x)$, $f_5(x)$ becomes a lot more probable than $f_6(x)$, etc as $x\to\infty$).

The summation $\sum_i i f_i(x)$ is now interpreted as the expectation of $X$, which, intuitively, must increase in $x$ since the probability mass is shifted towards larger values.