Beginner conceptual question about the relationship between probability density and distribution functions When introducing the relationship between a probability distribution function and a density function for a continuous RV, I see many books pointing this out:
$P(t<X<t+dt) = f(t)dt$
Questions


*

*In very very simple terms, why is this important to know? How would you explain this to a beginner amateur who hasn't taken calculus in 10 years?

*In practice would anyone ever use this for a calculation? For example let's say that $X$ is the amount of rain we'll get tomorrow in inches. I want to know the probability of getting 10 inches of rain tomorrow. Then I do $f(10)*dt$ which I'm guessing would equal zero and be useless (since a continuous RV has probability 0 for a specific value).
Note that the following relationship does make a lot more sense to me.
$\frac{d}{da}F(a)=f(a)$
 A: The identity $$\frac{d}{dt} F(t) = f(t)$$
might be in practice more useful that the identity
$$
P(t<X<t+dt) = f(t)dt
$$
since the first one gives you something you can compute.
But these two are formally equivalent. The notation $\frac{d}{dt}$ stands for comparing the variation of a function (say $F$) with respect to small variations of time. So
$$dF(t) =F(t + dt) - F(t)$$and thus formally
$$
f(t) = \frac{dF}{dt} = \frac{F(t + dt) - F(t)}{dt} = \frac{P(t<X<t+dt)}{dt}
$$
multiplying everything with $dt$ gives us the first formula.
The positive thing about the second formula is that 


*

*you can see the mathematically important thing happening: the probability of falling in a box becomes smaller as we reduce the box's size. This is not always the case, say if $X = 0$ is deterministic, then for any box around zero the probability of falling in that box is constantly $1$.

*you get a pictorial representation of your density function, since the formula tells you that you can approximate $f$ through a histogram.


For the mathematician the first reason is the most important one, and it suggests a change of paradigm. Whereas in analysis the derivative plays a central rôle, in modern probability theory it is integration that plays a central rôle. The idea is that a probability measure is a way of integrating. And the above formula tells you that this new way of integrating does not differ from the classical one. In some fancy notation one writes:
$$
P(s <X<t) = \int 1_{[s,t]}(X) dP = \int_s^t f(u) du.
$$
