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Problem 13 Medium Difficulty

$ REM $ sleep is the phase of sleep when most active dreaming occurs. In a study, the amount of $ REM $ sleep during the first four hours of sleep was described by a random variable $ T $ with probability density function
$$ f(t) = \left\{
\begin{array}{ll}
\frac{1}{1600} t & \mbox{if $ 0 \le t \le 40 $}\\
\frac{1}{20} - \frac{1}{1600} t & \mbox{if $ 40 < t \le 80 $ }\\
0 & \mbox{otherwise}
\end{array} \right.$$
where $ t $ is measured in minutes.
(a) What is the probability that the amount of $ REM $ sleep is between 30 and 60 minutes?
(b) Find the mean amount of $ REM $ sleep

Answer

A. $\approx 59.4 \%$
B. 40 $\mathrm{min}$

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Video Transcript

So the question asked us to You find the round sweet between 30 and 16. And the problem gives us these parameters Here, this first function, if we are in grams, sleep greater or equal to zero minutes and the last nickel to 40 minutes and this second function if were greater than 40 minutes but also best than or equal t 80 minutes and zero if it does not fulfill any of these two time requirements. Great. So her a is asking us to find the room sleep between 30 to 60 minutes. And with that, we need to use both functions that were provided above because together both functions with Phil this 32. 16 in that time period. So to start off, the are going to have our definite integral with their limits 30 to 40 has given in the first function above with one over 1600 t d t. Don't forget your d t. That is what helps you actually and agree. And our second portion would be 40 to 60 minutes with the function 1/20 minus one over 1600 t Bt Wonderful. So we have our in a rolls and our limits set up. Do you make this a little easier? And since this is a continent or a constant, we're going to pull out this one over 16 hundreds that will have one over 1600 and then we can actually integrate. That's he right above. And remember, this is technically t to the first. So integrated. We want to add one to the exponents to give us tea scored and then divide by whatever this new experiment is. So it happens to meet, too, and we want to keep her limits 30 2 40 And then for our second portion, we have one over 20 T minus one over 1600 similar to the first portion t squared. We were too under limits, 40 to 60. So that's what we're working with. We are now done with a calculus portion, and what I like to consider the hard portion is the algebra, because this is where you can start to make all of these mistakes. Um, like plugging in an order of operations. So we just want to keep that in mind as we are doing problem. So to start off will keep this one over 1600 and we're going to just straight substitute in. We have 40 squared over to my nest, 30 squared over to. And if this is a little confusing to see, this is the same thing is 1/2 over T square just were in in a different way. So that's our first question. And our second question is going to be 1/20 times 60 minus one over 1600 times 60 squared. Well, Rick, too. Oh, this deceits, uh, minus one. Over 20 trains, 40 minus one over 1600 times 40 squared over to really long and complicated re, and you can see why algebra can get a little confusing, a little messy. We're gonna break this all down. So when we simplify this first portion here we are going to get wait to one, eat no. 218 75 And then when you simplify Oh, this will get plus 1.875 That's what you get when we simplify the green minus 1.5. And that's what he checked out when we simplified blue. So now we're street. Just, uh, adding and subtracting. So looking at 0.21875 Let's 0.375 and that'll give us Wait, My nine, 375 Remember, this is a study, and when you give someone decimals a little hard to understand, so for practicality, senses will multiply this by 100%. And we'll get a final answer when we round 2 59 Wait for person. And that's the answer Report. A heart beats an ass Us. Do you find the me amount of friends Week? And to do that, we want to add Hall of our non zero parameters, meaning that first in section election and in my medical senses mean his reign as Mu and we want to add all our non zero portions rate. So they give it to infinity to infinity. The T deserve ill variable and lift functions. And for tea DT mathematically, I is what is written. Hey, logistically, it doesn't make too much sense, right? So we're gonna start putting in our functions. Teoh hopefully and hopefully that I don't need a little more sense. So Oliver, non zero parameters rates So that first function gave us limits of 0 to 40 t as oh, that's a variable. We're trying to find times one over 1600 t day 80 plus you want to add our second function, which had the limits 42 et and T That's variable that we're trying to find. And then the function of they gave us the sunrise. That was 1/20 minus one over 1600 t And of course, Haouari 18 and again. So we have this constant here so we can pull this constant out. So that's what Would you have One wish? 1600. And now, um, way can multiply this tea in this team together which will give us t squared. I can t well, this our 40 t 80 and then we can distribute this t tow both terms, so we'll have 40 to 80 in our limits of 1/20 T minus one over 1600 t squared Don t which we look at, It can look a little like some hard. Um, remember, we're gonna break this time piece by piece, sailor. First step is to an agreed. We're going to keep our one over 1600 on the outside and against integrate, we want to add one to her exponents and then defined by that new exponents, so t cute over three over here. Zero t 40 plus our second inter goal. Well, our second integration, which would be 1/20 but t squared over to minus one over 1600 times t cute over three. 46 42. You all right? And hopefully this starts to make a little more sense. So we're gonna keep her one over 1600 on the outside. Appear we can put our 40 cubes over three minus zero. Keen over three. And that's just zero. Plus our 1/20 times 80 squared over two minus one over 1600 times. Any cute over three minus. We went over 20 chains 40 squared over two minus one over 1600 times. 40 kids over three. And again, this is where algebra, uh, can get a little hard. Little tricky if not careful if you move too fast by with that first, this first funeral here, this simplifies to 13.33 roughly. And then this second portion here. Ah, when we do Oh, we'll get it. Plus 53 made 33 minus 26 6 seven. And again, we want to subtract this and then at that and then antis 13.33 So when we simplify that, we get about 39.96 minutes or to round this up and make it nice and me about 40 minutes.