Component follows the failure frequency / failure intensity function $$\lambda(t) = \kappa \beta t^{\beta-1} ,\;\;\kappa,\beta\in \mathbb{R}$$ What is the probability that the component fails during interval $t\in[a,b]$. The component cannot be repaired.
My initial idea was to use non-homogeneous Poisson point process.
$$ P\{N(a,b]=n\}=\frac{[\int_a^b \kappa \beta t^{\beta-1} dt]^{n}}{n!}e^{-\int_a^b \kappa \beta t^{\beta-1} dt}. $$
So I thought the probability of failure (at least one) between the interval is the complement of an event that there is no failure during that interval.
\begin{align*}P\{N(a,b]> 0)\} &= 1- P\{N(a,b]=0\} \\ &= 1-e^{-\int_a^b \kappa \beta t^{\beta-1} dt}\end{align*}
But this would mean that the total probability of that interval is 1 which doesn't make any sense. How should I approach this problem?
EDIT: I made mistake of thinking the total probability of the interval is 1, but i'm still not convinced my approach is the correct one.
I would have thought that if the hazard function is $\lambda(t) = \kappa \beta t^{\beta-1}$
then the cumulative hazard function is $\Lambda(t) = \displaystyle \int_0^t \lambda(u)\, du = \kappa t^{\beta}$
and the survival function is $S(t) = \exp(-\Lambda(t)) = \exp(-\kappa t^{\beta})$
making the probability of failure in $[a,b]$ be $S(a)-S(b) = \exp(-\kappa a^{\beta}) - \exp(-\kappa b^{\beta}) $
which tends to $1$ when $a \to 0, b \to \infty$