Let $p\_i \in [0,1]$ and $\sum\_{i} p\_i = 1$, and furthermore let $a\_i$ and $b\_i$ be positive real numbers. Is the inequality
$$
\sum\_{i} p\_i \frac{a\_i}{b\_i} \leq \frac{\sum\_{i} p\_i a\_i}{\sum\_{i} p\_i b\_i}$$
true?



Is the inequality $\sum_{i} p_i \frac{a_i}{b_i} \leq \frac{\sum_{i} p_i a_i}{\sum_{i} p_i b_i}$ true?
**No**. This reads
$${\mathbb E}[f(a,b)]\le f({\mathbb E}[a,b])$$
for an arbitrary discrete probability. This is true if and only if $f$ is a concave function (Jenssen Inequality). But $f(x,y)=\frac xy$ is not concave (nor convex).


You might prefer the convex function $g(x,y)=\frac{x^2}y$, for which we thus have
$${\mathbb E}\left[\frac{a^2}b\right]\ge \frac{{\mathbb E}[a]^2}{{\mathbb E}[b]}.$$


