Minimize the Sum of Reciprocal of Probabilities I need a probability distribution over $ n $ events such that the sum of expected values is minimized. That is, minimize 
The problem is given by:
$$\begin{aligned}
\arg \min_{ {p}_{i} } & \; && \sum_{i = 1}^{n} \frac{1}{ {p}_{i} } \\ 
\text{subject to} & \; && \sum_{i = 1}^{n} {p}_{i} = 1
\end{aligned}$$
I guess it's a basic easy question. I would appreciate any hint.

Here is an argument that tells you that the minimum value can only be  attained the $p_i$'s are equal. Suppose the minimum value is attained when $p_i=q_i, 1 \leq i \leq n$. If possible let $q_i \neq q_j$. Note that $\frac 1 {q_i} +\frac 1 {q_j} >\frac 1 {\frac {q_i+q_j} 2}+\frac 1 {\frac {q_i+q_j} 2}$. [This is simply a re-writing of the inequality $(q_i-q_j)^{2} >0]$. Thus there is a choice of $p_i$'s for which we get a value lower than the minimum. This contradiction shows that $p_i$ are all equal when the minimum is attained. 
