My work to this point is as follows: If $x > 0$, then [a theorem from my book] tells us that

\begin{align*}
\int_\mathbb{R} F(x) \, d\mu_F(x) &= \int_\mathbb{R} \mu_F \bigl((0, x]\bigr) \, d\mu_F(x)\\
&= \int_{\mathbb{R}} \int_{\mathbb{R}} \boldsymbol \chi_{\{ y \in \mathbb{R} \colon y \in (0, x] \}} \, d\mu_F(y) \, d\mu_F(x)\\
&=
\end{align*}

what if you want leaves from the company you work at for a certain duration which will be say 1 month from now and the company promotes you and your new job location is far from the place you were posted at earlier, but you have a fear that at new location they won't allow you leaves for that certain duration?
in such a case, will you resign from the company or wait to see whether they will give you the leaves you want? The latter has almost zero chances .

@TedShifrin Which problem set is this question "D" on that you're talking about here? https://youtu.be/l8rqK4sJuP8?t=2249

I have all the p-sets so just need a number.

\begin{align}
\mu(A) &= \int_{\mathbb{X}_1} \mu_{x_1}(A_{x_1}) \, d\mu_1(x_1) \notag \\
&= \int_{\mathbb{X}_1} \left(\int_{A_{x_1}} \frac{1}{\Gamma(2)x_1^2}x_2^{2 - 1} e^{-\frac{x_2}{x_1}} \, d\mu_1(x_2)\right) \, d\mu_1(x_1)
\end{align}

\begin{align*}
\mu_{\theta}(A_{\theta}) &= \int_{A_2} \frac{1}{\Gamma(2)\theta^2}x_2^{2 - 1} e^{-\frac{x_2}{\theta}} \, d\mu_1(x_2)
\end{align*}