00:01
The time in minutes until the next busty parts follows a distribution with the density function shown here.
00:09
And for part 1, we want to find the probability that the time is at most 35 minutes.
00:16
So this is the probability that x is less than equal to 35.
00:22
And so we are asked to sketch and label a graph of the distribution.
00:32
So we can see that the density function is constant at 1 over 20.
00:36
This is known as a uniform distribution, and it exists or is non -zero for only x between 25 and 45.
00:53
So suppose x equals 25 is here, and x equals 45 is here, and the height of 1 over 20 is here.
01:11
This density function makes a rectangle, and 35 would be here, so the probability that x is bigger than 35 is equal to the area under the curve between 35.
01:27
And 45.
01:31
So this probability can be solved by integrating the density function from 35 up to 45.
01:41
45 is the maximum value and this comes out to 1 half.
01:58
And then for part 2 we want to find the probability that the time is between 35 and 40 minutes.
02:10
So i'm just going to reuse the graph from question 1 and modify it.
02:21
So 40 is here.
02:27
So the probability that x is between 35 and 40.
02:31
It's equal to the area under the curve between these two numbers.
02:54
And this comes out to 1 over 4.
02:58
So you can also note that the waiting times are a uniform random variable on 25 to 45.
03:13
And then for part 3, we want the probability that x is between 25 and 55.
03:33
Now x is only non -zero up to 45, so we can only integrate up to 45.
03:40
And so this is an integration over the entire density function, so this comes out to 1.
03:53
And then for 4 we want to find the 90th percentile.
03:58
Now the 90th percentile is by definition the value of the random variable such that 90 percent of the distribution is below it...