TITLE: Derivatives through random variables?
QUESTION [5 upvotes]: Suppose I have some random variable X with probability distribution P(.;theta). Suppose I have a single sample x from this distribution.
Does it make any sense to ask for the derivative of x with respect to theta?
I believe it does not, but am having trouble convincing some of my colleagues.

REPLY [5 votes]: What might make sense is if $X$ is a function $g(U,\theta)$ of some underlying random variable $U$ (with distribution not depending on $\theta$) and $\theta$, where the differentiable function $g$ is chosen so that $g(U,\theta)$ has the given (continuous) distribution.  Then you could say $\dfrac{dX}{d\theta} = \dfrac{\partial g}{\partial \theta}(U,\theta)$.  Of course this depends on the "implementation", i.e. the choice of $g$ and $U$, rather than just on the distribution of $X$.