Suppose that the lifetime of a device is exponential with rate λ, but suppose also that the value of λ is not fixed but is itself a random variable that is uniform in the range [a, b) with 0 < a.
Can the pdf of the lifetime of the device be written as:
$ \lambda = \begin{cases} 0, x < a\\ \dfrac {1} {b-a} , a\leq x < b\\ 0, x\geq b \end{cases} $
Let $T|\lambda \sim \text{Exp}(\lambda)$ be the lifetime of the device and let $\lambda \sim \text{U}[a,b)$ be the parameter. Then for all $t \geqslant 0$ you have:
$$\begin{equation} \begin{aligned} f_T(t) &= \int \limits_a^b p(T=t|\lambda) \pi(\lambda) d\lambda \\[6pt] &= \frac{1}{a-b} \int \limits_a^b \exp(-\lambda t) d\lambda \\[6pt] &= \frac{1}{a-b} \Bigg[ - \frac{1}{t} \cdot \exp(-\lambda t) \Bigg]_{\lambda=a}^{\lambda=b} \\[6pt] &= \frac{1}{a-b} \Bigg[ \frac{\exp(-at) - \exp(-bt)}{t} \Bigg] \\[6pt] &= \frac{e^{-at} - e^{-bt}}{a-b} \cdot \frac{1}{t}. \\[6pt] \end{aligned} \end{equation}$$