# Exponential distribution as a differential equation

I'm trying to interpret the following situation. In an economy, let $T$ denote the remaining lifetime (a stochastic variable) with exponential distribution and a Cumulative distribution function satisfying the following differential equation:

$$F'(t) = (1- F(t))p$$.

I would like to interpret this equation and the parameter p.

My attempt is to observe that the instantaneous rate of change of the cumulative distribution; i.e., how rapidly the probability of observing $T≤t$ is increasing, is $F'(t)$. With this, I think that this rate is iqual to the probability $P(T>t)$ weighted by $p$. But this is not convincing.

And how about $p$?

• Hint: $$-p = -F^\prime(t)/(1-F(t)) = \frac{d}{dt}\log(1-F(t))$$ is the hazard function. – whuber Mar 13 '18 at 14:58

The differential equation $$F'(t) = (1-F(t))p$$ has general solution $$F(t) = 1 + Ce^{-pt}$$. Now, since $$F$$ is a nonnegative distribution, $$F(0)=0$$, and hence $$C=-1$$. So the distribution is $$F(t) = 1-e^{-pt}$$, an exponential distribution with parameter $$p$$.