Partial derivatives of random variables

Question

I have a question about partial derivatives and random variables, say we have two random variables, $r$ and $s$, and they are related by the expression $s=e^r$. Now, $r$ has an explicit dependency on the variable $t$, $r(t)$. Hence, $s$ doesn't explicitly depend on the variable t. Say we want to compute $\partial s/\partial t$. Can I just use the chain rule? I have seen some arguing that this is zero, but I'm not convinced. Note that I want the partial derivative, not the total derivative. Thank you.

Answer 1

I think the confusion comes from how you define the partial derivative. Usually, in calculus books, you have something like $z=f(x,y), x=g(u,v),$ and $y=h(u,v)$, and you can use the chain rule to find $\partial z/\partial u$. When you do, the definition of the “partial” derivative is that $v$ is held constant. You would NOT define $\partial z/\partial u$ as holding $x$ and $y$ constant.

However, in physics we often have a quantity that depends on position and time, with position being a function of time, say $f(x(t), t)$. If for a particular function $f$ there was no explicit time dependence, then we say $\partial f/\partial t =0$; we wouldn’t use the chain rule with $x$ because we’d be holding $x$ constant.

I’m not sure which is more appropriate for your situation.

I should add that I’m not sure how “random variables” factors into your question. $r$ seems to be a deterministic function of $t$.