19. A crate with weight is pushed against the wall by a force . The coefficient of static friction between the crate and wall is . Determine the probability that the crate will slip. P and W are distributed independently.

Solution

Assume no movement.

From above two equations, we have

Thus, the distribution of the frictional force is and the distribution of the maximum frictional force is .

Then we construct function

Thus, the probability that the crate will slip is

Ans.