The traffic light is green for pedestrians for 36 seconds and for cars for 24 seconds. Zhenya can cross the street in 6 seconds. If the traffic light changed color while Zhenya was crossing the road, then Zhenya increases his speed by 3 times. If Zhenya approaches the traffic light, and it is lit red, then he waits until the color changes to green. Zhenya approaches the traffic light at a random moment in time. Find the mathematical expectation of the time after which he
will cross the road (including the waiting time).

You want the expected time it takes for Zhenya to get across a street if it normally takes him 6 seconds, but he walks 3 times as fast when the light changes while he is in transit. If he arrives at the crossing when the light is red, he waits. The light is green for 36 seconds, and red for 24 seconds.

Crossing time

If Zhenya's arrival time is within the 24 seconds the light is red, the crossing time is 6 seconds plus the wait time.

If Zhenya's arrival time is within the first 30 seconds the light is green, the crossing time is 6 seconds.

If Zhenya's arrival time is within the last 6 seconds of the green period, his crossing time is 2 seconds plus 2/3 of the remaining green period.

The crossing time function is shown in the attachment.

Expected value

The expected value of the crossing time is the average value of the crossing time function. That is found by integrating the function over the range of arrival times, and dividing the integral by that range.

The expected value of crossing time is 10.6 seconds.