18. A force  acts on a crate whose weight is .  The coefficient of static friction between the crate and floor is . Determine the probability that the crate will move forward. P and W are distributed independently.

Solution

Assume no movement.

From above two equations, we have

The maximum frictional force is

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 move forward is

Ans.