This is a question from a textbook titled Probability and Stochastic Processes by Leo Breiman. This is problem 12 of Chapter I of the book.
A current is switched on sometime "at random" during a day. Then it is switched off "at random" between the time current is switched on and the end of the day. Let the outcome of the experiments be these two times.
Find appropriate density function f(x,y)?
For this part, I assumed it to be a uniform density function and found its value of it by integrating over the region $$ 0 \le x \le 24 \ {\rm and} \ x \le y \le 24 $$
$$ \iint_A c \,dx\,dy $$
where A is the interval given above.
Evaluate the probability that the current was on at least half the day.
For the second part, I evaluated integral over $$ 12 \le x \le 24 \ {\rm and} \ x \le y \le 24 $$.
Is my approach correct because my answer is far from what is mentioned in the book?