Having $T_C$, as a constant, as well as $m$ different constants ($\lambda_1, \lambda_2, \lambda_3,\dots, \lambda_m$), we declare the following probability for the $i^{th}$ event to happen: $p_i(T_C,\lambda_i)=1-e^{-\lambda_i T_C}$ .
Now, we need to change $\lambda_i$ to be a function of time as follows: $\lambda_i(t)=V_i\beta(t-\tau)$ in which $\tau$ is constant. We assume that $V_i (1\le i \le m)$ form an i.i.d. sequence of random variables distributed around some reference $V$.
We also denote $\phi^{}_{V}(x)=E(e^{tV})$ to be the moment generating function of $V$ and $\phi^{′}_V (x)$ its first derivative.
The $\beta$ function has the following characteristics:
- $\beta(t) \ge 0\ \forall t$ with $\beta(0^+)>0$,
- $\beta(t) = 0\ \forall t < 0$,
- $\beta(t)$ continuous almost everywhere,
- $\int_0^{\infty}\beta(t)dt = 1$.
The question is that what $p_i(T_C,\lambda_i(t))$ would look like?
I have the solution but I don't know how we could get to this solution?
The solution:
$p_i(T_C,\lambda_i(t)) = 1 - \int_0^{\infty} \beta(\tau)\frac{\phi^{′}_V \big(-\int_0^{T_C}\beta(\tau - \theta)d\theta\big)}{E(V)}d\tau$