What method would I use to find the PDF of a random variable that has a parameter as a realisation of another random variable?
For example, I first have an exponential distribution $\Omega \sim exp(\lambda)$ which has a realisation of $\omega$.
Then I have another normal distribution $\mathcal{T} \sim \mathcal{N}(\omega, \sigma^2)$ (i.e. the mean of $\mathcal{T}$ is the realisation $\omega$).
I am trying to find the PDF of the second distribution $\mathcal{T}$. It would be helpful if I could understand the general method used here because I am also trying to find the PDF of other continuous distributions that use the realisation $\omega$ as a parameter.
Thank you
Independence of the normal and the exponential is needed for computing the density function. In the present case the density function is $\frac 1 {\sqrt {2\pi}\sigma}\int e^{-(x-a)^{2}/2\sigma ^{2}} \lambda e^{-\lambda a}da$. I hope the general procedure is clear from this formula. Of course the PDF is obtained by integrating the density function from $-\infty$ to $x$.