Recently I had to calculate percentage increase in exponential cdf in an interval of length 1. So:
$\frac{F(x + 1) - F(x)}{F(x)} = \frac{(1 - \exp(-\lambda(x+1)) - 1 + \exp(-\lambda x)}{1 - \exp(-\lambda x)} = \frac{\exp(-\lambda x)(1 - \exp(-\lambda))}{1 - \exp(-\lambda x)}$,
therefore the value depends on $x$. However, when calculating decrease in $1-F(x)$ (which is a definition of survival function), we have:
$\frac{S(x + 1) - S(x)}{S(x)} = \frac{\exp(-\lambda(x+1)) - \exp(-\lambda x)}{\exp(-\lambda x)} = \frac{\exp(-\lambda x)(\exp(-\lambda)-1)}{\exp(-\lambda x)} = \exp(-\lambda)-1$,
so the percentage decrease is constant. Here's my question: am I doing some mistake in calculation? The fact that a percentage increase of cdf is not constant, but the percentage decrease of survival function IS constant seems pretty counterintuitive to me.
I think your calculation looks correct (remember it is negative) and what you have observed is fairly intuitive:
The constant percentage decrease in the survival function of an exponential distribution is a result of the memoryless property: the conditional probability decrease is therefore constant over time and so the actual decrease in the probability of survival over the next period is proportional to the probability of surviving so far
But the probability of survival so far reduces over time, so the actual decrease in the probability of survival over the next period also reduces over time. At the same time, the probability of failing to survive so far (the cumulative distribution function) is increasing over time, so the ratio of these two must decrease over time