I have a general question about survival functions and their associated PDFs (probability density functions).
Background. A survival function $s(x)$ is the probability that an individual will survive $x$ more years. In particular $s(x) = P(X > x)$. Also $\mu(x)$ is the "force of mortality." In particular, $\mu(x) \ dx$ gives the probability that an individual aged $x$ will die in the interval $(x, x+dx)$.
Question. Why is the pdf of $X$ the following: $f_{X}(x) = s(x) \cdot \mu(x)$ where $\mu(x)$ is the "force of mortality"?
My Answer. The random amount of years $X$ a person will live is conditional upon the fact that he survived up to the current time? So we have to weight the fact that the person has probability of dying or living at some time?