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 $p$?

  • 3
    $\begingroup$ Hint: $$-p = -F^\prime(t)/(1-F(t)) = \frac{d}{dt}\log(1-F(t))$$ is the hazard function. $\endgroup$ – 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 $p$.


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