Deriving the density function of a exponential distribution

Question

I am reading a book where the author tries to derive the density function of a exponential variable by the following form:

Suppose that in a short interval of time $\Delta t$ there is a chance $\lambda \Delta t$ that a event will occur. assuming $\Delta t$ is very small, the chance of more than one event will occur is negligible so that the chance that no event occurs is $(1-\lambda \Delta t)$. It will be assumed that the chance of an event occurring in a short time does not depend on how many events have already occurred.

Is we write $F(t)$ for te probability that no event occurs before time $t$, then the probability that no event occurs before time $F(t+\Delta > t)$ if the probability that no event occurs before time $t$ multiplied by the probability that no event occurs between time $t$ and time $(t > + \Delta t)$; in symbols ,

$$F(t + \Delta t)= F(t)[1-\lambda \Delta t]$$

$$\frac{F(t + \Delta t) -F(t)}{\Delta t}= -\lambda$$

The author then goes on to use the definition of derivative to conclude the prove, which I fully understand. My problem is with the last two equations. For me there is some error since if we isolate $\lambda$ in the first equation we get

$$\frac{F(t + \Delta t) -F(t)}{F(t)\Delta t}= -\lambda$$

Can someone enlighten me where is my mistake?

Answer 1

You have made no mistake. The passage you quote should lead to the conclusion that $$\frac{F(t+\Delta t)-F(t)}{\Delta t}\approx -\lambda F(t).$$ (The first equation mentioned is more or less correct, though of course we do not have actual equality. The second equation is just plain wrong.)

Then taking the limit we find that $F'(t)=-\lambda F(t)$. This is the familiar DE of exponential decay, and we get that $F(t)=Ce^{-\lambda t}$ for some constant $C$ which turns out to be $1$.

Remark: The choice of $F(t)$ for the name of the probability that no event occurs before time $t$ is very unfortunate. That's because $F(t)$ in this case is $1$ minus the cumulative distribution function. The usual convention is that $F(t)$ is the name of the cdf.