00:02
We're told that waiting time for a bus in the morning is uniformly distributed on the interval 08, and waiting time in the evening is uniformly distributed on the interval 010, independent of the morning waiting time.
00:21
In part a, suppose that we take the bus each morning and evening for a week, we're asked to find our total expected waiting time.
00:39
So, first let's define some random variables.
00:44
So let x1 through x5, these five random variables be the morning times during this week, and x6 through x10 be the evening times.
01:14
We have that we want to find the total expected waiting time.
01:22
This is the expected value of the sum of x1 through x.
01:26
Because this is a linear combination this is the same as the sum of the expected values so expected value of x1 up to expected value of x10 now we have that all the morning week times are the same so they all have the value expected value of x1 so we have five times expected value of x1 and all the evening times are the same they have the same expected value so we have five times the expected value of x6 for the evening times.
02:18
And we're given the expected values for the morning times is four, so five times four, and the expected value for the evening time is five.
02:33
So we have five times five, so we have five times four plus five times five, which is 45.
02:45
So the expected total weight time is ten.
02:54
In part b, we're asked to find the variance of the total waiting time.
03:07
Well, we have the variance of x1 summed through x10.
03:16
This is the same as the sum of the variances, x1 up through the variance of x10.
03:27
And then, because we have the evening times or independent of the morning waiting times, then this is simply what the variance is.
03:55
And so this can be simplified since the morning times have the same variance to five times the variance of x1.
04:02
And the evening times also have the same variance plus five times the variance of x6.
04:10
And we're given that the variance is for x1 and x6.
04:22
These are calculated by squaring the length of the interval.
04:31
So we have 5 times 64 and dividing by 12.
04:56
And then we have squaring the other length.
05:01
This is 10 squared or 100.
05:04
Again, divided by 12.
05:17
And so we get 820 over 12, which is equal to 68 .33.
05:37
And this is our variance.
05:43
In part c, we're asked to find the expected value in variance of the difference between morning and evening waiting times on a given day...