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 $p$?