Exponential distribution - direct proportionality I just can't understand exponential distribution. Random variable $X$ has an exponential distribution if the probability
$$P(t < X \le t + \Delta t | X>t)$$
is (approximately) directly proportional to the length of time interval ($\Delta t$), the ratio of proportionality is the parameter of exponential distribution - $\lambda$.
So if I double $\Delta t$, the probability will be two times higher, right?
Now, suppose $P(t < X \le t + \Delta t | X>t)=0.3$. By doubling $\Delta t$, I should get the probability of $0.6$. Double it again, I get $1.2$. Oops! Probability should stay within range of $[0,1]$, right!
There's some issue in my reasoning I'm unable to spot. Could anyone help?
 A: Let $X$ be an exponential random variable.Then the pdf is 
$f(x) = \lambda e^{-\lambda x}; x \gt 0 $ and cdf is $F(x) = P(X \leq x) = 1 - e^{-\lambda x}; x \gt 0$. 
Suppose we know $X > t$, then what is the probability that $X > t + \Delta t$ for some $\Delta t \geq 0$. i.e. we want to know $P (X > t+\Delta t / X > t)$. Now,
\begin{equation}
\begin{split}
P (X > t+\Delta t / X > t)  &= \frac{P(X > t+\Delta t, X > t)}{P(X > t)} \\
&= \frac{P(X > t+\Delta t)}{P(X > t)} \\
&= \frac{1-(1-e^{-\lambda (t+\Delta t)})}{1-(1-e^{-\lambda t})} \\
&= e^{-\lambda \Delta t} \hspace{2ex} \text{(independent of t)}
\end{split}\label{1}\tag{1}
\end{equation}
\begin{equation}
\begin{split}
P (X < t+\Delta t / X > t)  &= 1 - P (X < t+\Delta t / X > t) \\
&= 1 - e^{-\lambda \Delta t} \hspace{2ex} \text{(exponential with param $\lambda$ )}
\end{split}
\end{equation}
It turns out that the conditional probability does not depend on $t$. The probability of an exponential r.v exceeding the value $t+\Delta t$ given it already exceeded $t$ is same as the variable originally exceeding $\Delta t$ regardless of $t$. This is called memoryless property since the past has no effect on it's future. Every instant is like the beginning of a new random period, which has the same distribution regardless of how much time has already elapsed. 
So, to answer your question, the conditional probability is not directly proportional to $\Delta t$, it is exponentially related to $\Delta t$. Since the conditional distribution also follows the exponential distribution, probability can never be greater than 1.
