I have two random variables, $X$ and $Y$, that have the distributions $$X \sim \beta(r_1 = 1, s_1 = 5) \qquad \text{and} \qquad Y \sim \beta(r_2 = 3, s_2 = 1)$$ I need to find $P(X < Y)$.
I thought about using a double-integral, then using $\ f_X(x)\,P(Y > x)\,\mathrm{d}x$ to solve the problem.
Is there an easier way to do it?
No. That's it. $$\begin{align} \operatorname{\bf P}(X<Y) & = \int_0^1 f_Y(y)\int_0^y f_X(x)\operatorname{d}x\operatorname{d}y \\ & = \int_0^1 \frac{y^{\alpha_y-1}(1-y)^{\beta_y-1}}{B(\alpha_y,\,\beta_y)} I_y(\alpha_x,\beta_x)\operatorname{d}y \\ & = \int_0^1 \frac{y^{2}}{B(3,\,1)}\frac{ B(y;1,5)}{B(1,5)}\operatorname{d}y \\ & = \int_0^1 \frac{y^{2}}{\frac 1 3}\frac{ \tfrac 1 5 (1-(1-y)^5)}{\tfrac 1 5}\operatorname{d}y \\ & = \int_0^1 3 y^7-15 y^6+30 y^5-30 y^4+15 y^3\operatorname{d}y \\ & = \tfrac 3 8 1^8-\tfrac {15} 7 1^7+5 1^6-6 1^5+\tfrac {15} 4 1^4 \\ & = \frac{55}{56} \end{align}$$
PS: That is assuming the R.V. are independent.