Let $X_1, X_2$ and $X_3$ three independent random variables with pdf and cdf $f_{X_i}(x_i)$ and $F_{X_i}(x_i)$ for $x_i\geq 0$, receptively. Using CDFs and PDFs of $X_i$, find the probability $$P\{X_1<X_3<X_2<\alpha-X_3\}.$$
Is it $$ P\{X_1<X_3<X_2<\alpha-X_3\} \\ =\int_{x_3=0}^{\infty} \int_{x_2=0}^{\alpha-x_3} \int_{x_3=0}^{x_2} \int_{x_1=0}^{x_3} f_{X_1}(x_1)\, \mathrm{d}x_1 f_{X_3}(x_3)\, \mathrm{d}x_3 f_{X_2}(x_2)\, \mathrm{d}x_2 f_{X_3}(x_3)\, \mathrm{d}x_3~, $$ or $$ P\{X_1<X_3<X_2<\alpha-X_3\}= \int_{x_3=0}^{\infty} \int_{x_2=0}^{\alpha-x_3} \int_{x_1=0}^{x_3} f_{X_1}(x_1) \,\mathrm{d} x_1 f_{X_2}(x_2) \,\mathrm{d} x_2 f_{X_3}(x_3) \,\mathrm{d} x_3 ~? $$ Thanks.