# Expected value of ratio of two function of the same random variable

Let $$X$$ be a r.v. with absolutely continuous distribution and continuous strictly positive density $$f: \mathbb{R} \rightarrow [0, \infty)$$ and let $$g$$ a further given continuous density function.

Set $$Y = \frac{g(X)}{f(X)}.$$ What can we say about the $$\mathbb{E}Y$$?

If $$X\sim f(x)$$, then [since $$Y$$ is a transform of $$X$$, by the law of the unconscious statistician] $$𝔼^Y[Y]=𝔼^X[\frac{g(X)}{f(X)}=∫\frac{g(x)}{f(x)}f(x)\text{d}x$$ which simplifies into $$∫_{\text{supp}(f)}g(x)\,\frac{f(x)}{f(x)}\text{d}x=∫_{\text{supp}(f)}g(x)\,\text{d}x$$ where $$\text{supp}(f)$$ denotes the support of $$f$$, that is the closure of the collection of points $$x$$ when $$f(x)>0$$. In the event when $$\text{supp}(g)\subset\text{supp}(f)$$ the integral is equal to one.