I have random variable (RV) $X$ where $X=\min\{X_1, X_1\cdot X_2\}$. Further, $X_1$ and $X_2$ are independent but not identical RVs with exponential distributions, i.e., $f_{X_1}(x_1)=\frac{1}{m_1}e^{-\frac{x_1}{m_1}}$ and $f_{X_2}(x_2)=\frac{1}{m_2}e^{-\frac{x_2}{m_2}}$.
I want to find the CDF of $X$, and I start as follows, but I am not sure how I can write following two terms in integral forms:
$$F_X(x) = \mathsf P(X < x) = \mathsf P(X_1 < x \mid X_2 \geqslant 1) + \mathsf P(X_1\cdot X_2 < x \mid X_2 < 1).$$
Can some one help me to write $F_X(x)$ in integral form using $X_1$ and $X_2$ PDFs/CDFs?
You were close.
$\begin{align}F_X (x) & =\mathsf P(X\leq x) \\ & = \mathsf P(\min\{X_1, X_1X_2\}\leq x) \\ & = \mathsf P(X_1X_2 \leq x\cap X_2 \leq 1)+\mathsf P(X_1 <x\cap X_2 > 1) \\ & = \int_{0}^{1} \int_0^{x/t}\frac {\mathrm e^{-t/m_2-s/m_1}}{m_1m_2}\operatorname d s\operatorname d t+\int_1^\infty\int_0^{x}\frac {\mathrm e^{-t/m_2-s/m_1}}{m_1m_2}\operatorname d s\operatorname d t \end{align}$