Example: Express d.f. of $Y = \min \{X1, X2\}$ in terms of joint d.f. $$H(x_1,x_2) = P(X_1 \le x_1, X_2 \le x_2)$$
In this exercise, they have an answer $P(Y \le x)= 1 - H(x,\infty) - H(\infty, x) + H(x,x) $ With $$H(x,\infty)=P(X_1 \le x,X_2 \le \infty)=P(X_1 \le x)$$
$$H(\infty ,x)=P(X_2 \le x)$$
What I did: I thought that it would be more realist to write that:
$$P(Y \le x)= 1 - P(Y > x)= 1 - P(X_1>x,X_2>x) = 1 - ( 1 - P(X_1 \le x_1, X_2 \le x_2))$$ and after that I could draw a little line with different postision of $X_1$ $X_2$ and then derive this formula: $P(Y \le x)= 1 + H(x,\infty) + H(\infty, x) - H(x,x) $
I can't see why they have : $P(Y \le x)= 1 - H(x,\infty) - H(\infty, x) + H(x,x) $