I understand the derivation leading to the memoryless property, but I cant seem to understand its application in a real scenario.
Let say we have a period $P$, in which no event occurs, if the memoryless property is true, it seem to me that the probability that an event occur at any instance $i$ in $P$ is a constant.
The memoryless property is defined as: $$\mathbb{P}(T> s+t \vert T > s) = \mathbb{P}(T> t)$$ In case of that $t$ is approaching $0$ (i.e the very next instance). $$\mathbb{P}(T> s+0 \vert T > s) = \mathbb{P}(T> 0)$$
Meaning, the probability that an event occurs in the very next instance equal to the probability that an event occurs the very next moment after initial time. And you can do the same thing for every instance $i$ during $P$.
Meaning, during period $P$ in which no event occurs, the probability of an event occur in any instance $i \in P$ is equal to $\mathbb{P}(0)$.
If that is true, why even has a CDF or PDF, as the value is equal to the PDF at time $0$ anyway. This has been bugging me several days now, can someone explain it to me ?
For all $t > 0$, $$\Pr[T > s + t \mid T > s] = \Pr[T > t] \tag{1}$$ implies that as $t \to 0^+$, we simply have $$\Pr[T > s \mid T > s] = \Pr[T > 0] = 1,$$ which of course is trivially true. But your interpretation of what this means is not correct. $(1)$ is the conditional probability that, having waited at least $s$ units of time without seeing the event, the probability of having to wait $t$ more units of time to see it is equal to the unconditional probability of having to wait at least $t$ units of time. It doesn't say that you actually see the event in the next $t$ units of time; rather, it's saying you have to wait at least $t$ more units.
The concept you are referring to is known as the hazard rate or hazard function, which is the instantaneous likelihood of seeing the event at a given moment in time, given that you have not already seen it. In other words, it is $$h(t) = \lim_{\Delta t \to 0^+} \frac{\Pr[T \le t + \Delta t \mid T > t]}{\Delta t}. \tag{2}$$ The interpretation is natural: given the event has not yet occurred by time $t$, the probability we observe it in within the next $\Delta t$ time units is $\Pr[T \le t + \Delta t \mid T > t]$. We then divide this by $\Delta t$ because we want a rate, not just a probability (which would be $0$ as $\Delta t$ becomes vanishingly small). So the quotient in $(2)$ is describing the rate at which this conditional probability is changing at time $t$, and the faster it increases, the more likely we are to see the event occurring in the next instant. But we can also rewrite $(2)$ as $$h(t) = \lim_{\Delta t \to 0^+} \frac{\Pr[t < T \le t + \Delta t]}{\Delta t} \cdot \frac{1}{\Pr[T > t]} = \frac{f_T(t)}{1 - F_T(t)}. \tag{3}$$ That is to say, the hazard function is simply the probability density $f_T$ divided by the survival $1 - F_T$ of $T$.
We can calculate this for an exponential distribution explicitly: $$f_T(t) = \lambda e^{-\lambda t}, \quad \Pr[T > t] = e^{-\lambda t}, \tag{4}$$ hence $$h(t) = \lambda. \tag{5}$$ The fact that this is constant (does not depend on $t$) for an exponential distribution is also a consequence of memorylessness, since we can also use $(1)$ to obtain $(5)$: $$\begin{align} \Pr[T < t + \Delta t \mid T > t] &= 1 - \Pr[T > t + \Delta t \mid T > t] \\ &= 1 - \Pr[T > \Delta t] \\ &= \Pr[T \le \Delta t] \\ &= F_T(\Delta t). \tag{6} \end{align}$$ where we used memorylessness in the second step. Consequently, $$h(t) = \lim_{\Delta t \to 0^+} \frac{F_T(\Delta t)}{\Delta t} = F'_T(0) = f_T(0) = \lambda e^{-\lambda \cdot 0} = \lambda. \tag{7}$$ Once we saw that the expression $(6)$ does not depend on $t$, we were assured that $h(t)$ is a constant function of $t$; hence, memorylessness implies constant hazard rate.