If $f$ and $g$ are two probability density functions with same support. Do we have $$\int \frac{f}{g} dx \geq 1? $$

For example, in a discrete case, we have positive numbers $(p_1, \cdots, p_n), (q_1, \cdots, q_n)$ such that $\sum p_i =\sum q_i = 1.$ Does the following inequality hold? $$ \sum p_i/q_i \geq n? $$

If so, how to prove? any counterexample if not?

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    $\begingroup$ Think about this for a little bit: This would imply $\int \frac g f \, dx \ge 1$ as well, by the same reasoning. Does this actually seem reasonable? $\endgroup$ – user296602 Oct 18 '17 at 16:22
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    $\begingroup$ Also, there are counterexamples with $n = 2$.... $\endgroup$ – user296602 Oct 18 '17 at 16:23
  • $\begingroup$ What does this have to do with expectation? You are not taking the expectation of the ratio RV you're instead integrating the ratio of two PDFs on their common support. $\endgroup$ – Nap D. Lover Oct 18 '17 at 16:36
  • $\begingroup$ Thanks @user296602. you are right... $\endgroup$ – VincentHall Oct 18 '17 at 17:43

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