I'm working on the following problem:
"The probability that an insured individual will give rise to no claims next year is $e^{-\theta}$, where $\theta$ varies by individuals according to the density function $f_\Theta(\theta) = 25\theta e^{-5\theta}$. What is the probability that a randomly selected individual will give rise to no claims next year?"
My intuition was essentially to multiply the probability of a certain $\theta$ by the probability of no claims for that $\theta$, and I approached that by finding $\int_0^\infty25\theta e^{-5\theta}e^{-(25\theta e^{-5\theta})}$, which came out as $\approx 0.335824$. Is this the correct way to approach the problem?
Not quite.
Suppose $\theta$ is given then the probability of no claim is is $\exp(-\theta).$
\begin{align} Pr(\text{no claim}) &= \int_0^\infty Pr(\text{no claim}|\theta)f_\Theta(\theta) \, d\theta\\ &= \int_0^\infty \exp(-\theta) f_\Theta(\theta) \, d\theta \end{align}