Alice is in a car that is driving down a bumpy road she is...

Alice is in a car that is driving down a bumpy road. She is holding her coffee mug which contains 20 Oz of coffee. Each time the car hits a bump a portion of the coffee spills out. We can model the remaining portion as a uniform random variable between zero and whatever the cup contains. The car hits the first bump and X Oz of the original 20 Oz of coffee is left for her. The car hits a second bump and now Y Oz of the X Oz is left for her. (a) Find the joint PDF of X and Y , that is, fX,Y (x, y). (b) Use the total (iterated) expectation to calculate the expected amount of coffee she is left with after hitting the second bump, i,e, E[Y ].

Transcript

00:01 So in this question we have been given that there's annie's arrival time and alvi's arrival time.

00:07 So i'm taking x as annie's arrival time.

00:14 And we are taking y as alv's arrival time.

00:25 Okay.

00:26 And they've told us that x and y are uniformly distributed on the interval 5 .6.

00:39 Okay.

00:40 So to start with the first part what they have asked.

00:44 We've been given the joint pdf of x and y and based on that we're supposed to write down the marginal pdfs and the joint pdf.

00:54 So here i can say that the joint pdf can be of like x and y.

01:03 You can write down this as product of marginal pdfs of x and y.

01:15 And this we are saying because both x and y are independent events so since they are independent i'm going to use this formula and first write down the marginal pdf so first i'm writing for x so we say f of x is since it's a uniform distribution it is one and zero so it is one between the interval five to six as per given question and it is zero elsewhere and similarly for y also this will be 1 and 0 for 5 between y also between 5 and 6 and it is 0 otherwise so combining them we can write out the joint pdf that therefore the joint pdf is f of x comma y can be written as this is 1 for the conditions of x as x is between 5 and 6 and similarly y is also between 5 and 6 and it is 0 elsewhere.

02:42 So this we've written is the joint pdf of x and y.

02:46 Now next part is to calculate part b where we say that probability they both arrive between 515 and 545.

03:09 So to find this probability, we're going to write this in terms of x and y intervals, so which represents as a x is between 515 and 545.

03:27 Similarly, we have y also between 515 and 545.

03:35 Now since the time interval, we are converting this into a number.

03:39 So we can convert 515 into a number by taking this as 515 means it is 5 hours.

03:48 So in 2 we do 16 minutes to convert into minutes and plus this is 15 minutes, right? and we can divide this in total by 60 minutes to again convert it into a number.

04:00 So which actually gives me 5 .25.

04:03 And in a similar way, 5 .45 also let us convert.

04:06 So this is 5 hours into 60 minutes plus we have another 45 minutes.

04:15 And the whole thing is again divided by 60 minutes.

04:19 So in terms of numbers, this will now change to 5 .75...

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