Probability of $X$ > $Y$, given $X$, $Y$ being the maximum of distinct Normal Distribution Families

Question

Let $ \begin{matrix} X_i & \sim & N(\mu_X, \sigma_X^2) \\ Y_i & \sim & N(\mu_Y, \sigma_Y^2) \\ X & := & \max(X_i, i \in \{ 1,2, ..., n_X\}) \\ Y & := & \max(Y_i, i \in \{ 1,2, ..., n_Y\}) \end{matrix} $

What is $\mathbb{P}[X>Y]$, as a function of $( \mu_X, \sigma_X, n_X, \mu_Y, \sigma_Y, n_Y )$?

Answer 1

$Pr(X>Y)=\sum_{k=1}^{n_x}Pr(X_k>max\{Y_i\}, X_k=max\{X_i\})$

$Pr(X_k>max\{Y_i\}, X_k=max\{X_i\})=\int_{x_k}Pr(x_k>max\{Y_i\}, x_k=max\{X_i\})f_X(x_k)dx_k=\int_{x_k}Pr(Y_1<x_k)^{n_y}Pr(X_1\leq x_k)^{n_x-1}f_X(x_k)dx_k=\int_{x_k}\Phi(\frac{x_k-\mu_y}{\sigma_y})^{n_y}\Phi(\frac{x_k-\mu_x}{\sigma_x})^{n_x-1}f_X(x_k)dx_k$

Thus $Pr(X>Y)=\sum_{k=1}^{n_x}\int_{x_k}\Phi(\frac{x_k-\mu_y}{\sigma_y})^{n_y}\Phi(\frac{x_k-\mu_x}{\sigma_x})^{n_x-1}f_X(x_k)dx_k$, where $f_X()$ is the density of $X_i$ and will depend on $\mu_x,\sigma_x$.