Let$\ H(u_1,u_2)$ and$\ M(u_1,u_2)$ be bivarate CDF with standard uniform margins. What probability will be defined by the following integral (if any): $ \int\limits_{[0;1]^2} H(u_1,u_2)\;dM(u_1,u_2) = P(\cdots)\ ?$
Thanks in advance.
Let$\ H(u_1,u_2)$ and$\ M(u_1,u_2)$ be bivarate CDF with standard uniform margins. What probability will be defined by the following integral (if any): $ \int\limits_{[0;1]^2} H(u_1,u_2)\;dM(u_1,u_2) = P(\cdots)\ ?$
Thanks in advance.
Assume that $H$ is the CDF of $(X_1,X_2)$, that $M$ is the CDF of $(Y_1,Y_2)$, and that $(X_1,X_2)$ and $(Y_1,Y_2)$ are independent. Then, $ \int H(u_1,u_2)\mathrm dM(u_1,u_2)=\mathrm E(H(Y_1,Y_2))=\mathrm P(X_1\leqslant Y_1,X_2\leqslant Y_2). $