I am currently studying the textbook Modeling and analysis of stochastic systems, third edition, by Kulkarni. Chapter 5.1.1 Memoryless Property says the following:
We begin with the definition of the memoryless property.
Definition 5.2 A non-negative random variable $X$ is said to have the memoryless property if
$$P(X > s + t \mid X > s) = P(X > t), \ \ \ \ \ s, t \ge 0. \tag{5.3}$$
Theorem 5.1 Memoryless Property. A continuous non-negative random variable has memoryless property if and only if it is an $\exp(\lambda)$ random variable for some $\lambda > 0$.
Proof: We first show the "if" part. So suppose $X \sim \exp(\lambda)$ for some $\lambda > 0$. Then,
$$\begin{align} P(X > s + t \mid X > s) &= \dfrac{P(X > s + t, X > s)}{P(X > s)} \\ &= \dfrac{P(X > s + t)}{P(X > s)} = \dfrac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\ &= e^{-\lambda t} = P(X > t). \end{align}$$
Hence, by definition, $X$ has memoryless property. Next we show the "only if" part. So, let $X$ be a non-negative random variable with complementary cdf
$$F^c(x) = P(X > x), \ \ \ \ x \ge 0.$$
Then, from Equation 5.3, we must have
$$F^c(s + t) = F^c(s) F^c(t), \ \ \ \ s, t \ge 0.$$
How does Equation 5.3 imply that $F^c(s + t) = F^c(s) F^c(t), s, t \ge 0$?
Best Answer
Note that $$P(X > s+t \mid X > s) = \frac{P(X > s+t, X > s)}{P(X > s)} = \frac{P(X > s+t)}{P(X > s)} = \frac{F^c(s+t)}{F^c(s)}$$ and $$P(X > t) = F^c(t).$$