Constant hazard in survival analysis and random event occurence I read that in survival analysis, when the hazard function $$\lambda(t)=\underset { dt\rightarrow 0 }{ lim } \frac { Pr(t\le T<t+dt|T\ge t) }{ dt } =c$$ then the event occurrence is random and independent (memoryless). 
I can understand the concept and implications of monotonically increasing hazard function. However is there any theory or proof that when the  $\lambda(t)=c$ then the event occurrence is random and independent?
 A: If the positive random variable $T$ denotes the time of failure of a system 
with hazard rate $\lambda(t)$, then it is straightforward to show that the
cumulative probability distribution function of $T$ is given by
$$F_T(t) = 1 - \exp\left(-\int_0^t \lambda(\tau)\,\mathrm d\tau\right), 
~ t \geq 0.$$
Thus, if the hazard rate $\lambda(t)$ equals a constant $c$ for all $t \geq 0$, we have that
$$\begin{align}
F_T(t) &= 1 - \exp(-ct), & t \geq 0,\\
P\{T > t\} &= \exp(-ct), & t \geq 0,\\
f_t(t) = \frac{\mathrm d}{\mathrm dt}F_T(t) &= c\exp(-ct), & t \geq 0,
\end{align}$$
showing that $T$ is an exponential random variable with parameter $c$. 
As is well-known, exponential random variables are memoryless:
$$\begin{align}
P\{T > s+t \mid T > t\} 
&= \frac{P\{(T > s+t)\cap (T > t)\}}{P\{T > t\}}\\
&= \frac{P\{T > s+t\}}{P\{T > t\}}\\
&= \frac{\exp(-c(t+s))}{\exp(-ct)}\\
&= \exp(-cs)\\
&= P\{T > s\}\\
\end{align}$$
which is better understood as saying that a system that has survived
till time $t$ forgets how old it is, and so the probability that it
lasts for an additional time $s$ is the same as the probability of
a brand new system surviving till time $s$. Systems with 
linearly monotonically
increasing hazard rates that the OP refers to
very definitely do not have this memoryless property. 
A: Maybe I misunderstand your question, but it might help to just think of the graph of the hazard function, that is, it's a line with no slope.  When the hazard is constant, there is no dependence on time and the instantaneous failure rate is the same at say, $t = 2$ as $t = 200$.  The exponential distribution has constant hazard, which makes sense when you think about the memoryless property of that distribution.
