Consider the following function:

$Y(t) = X_1 \exp{(-X_2 t)}$

where the parameters $X_1, X_2$ are random variables and $t$ is the time.

How to find the analytical expression of the probability density function of the function $Y(t)$?

  • 2
    $\begingroup$ Using just the definition, it's straightforward to write down an expression for the PDF--but whether you consider that "analytical," or even useful, is another matter. If you want more than such a generality, could you indicate what the distribution function of $(X_1, X_2)$ is? $\endgroup$
    – whuber
    Mar 15 '16 at 17:21
  • $\begingroup$ In a first time, we can suppose that random variables $(X_1,X_2)$ follow normal distributions: $X_1 \sim \mathcal{N}(m_1,\sigma_1)$, $X_2 \sim \mathcal{N}(m_2,\sigma_2)$ $\endgroup$
    – user41037
    Mar 16 '16 at 15:22
  • 1
    $\begingroup$ Some closely related posts which might also be helpful can be found by searching our site for normal lognormal product. $\endgroup$
    – whuber
    Mar 16 '16 at 16:54