Problem 12 Medium Difficulty

The time between infection and the display of symptoms for streptococcal sore throat is a random variable whose probability density function can be approximated by $ f(t) = \frac{1}{15,676} t^2 e^{-0.05t} $ if $ 0 \le t \le 150 $ and $ f(t) = 0 $ otherwise ($ t $ measured in hours).
(a) What is the probability that an infected patient will display symptoms within the first 48 hours?
(b) What is the probability that an infected patient will not display symptoms until after 36 hours?

Answer

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Video Transcript

Okay, So for this question we have, that team is defined to be won over 15 6 76 t squared you to be mine. A 0.5 teen. And this is true. Call t A Greely equals zero on dhe. Ever till these, you know, faulty zero. So the question is, we need to find the probability that infected patient will display symptoms within the 1st 48 hours. So, mathematically, this is the same. Must answer the question. What is P of tea lesson? Of course. What he ate. So turning this into into go, it becomes enter. Go from 0 to 48 o ff t t. So once you put ff t as we've seen already. So this is gonna be one over 15 676 on into go beauty. 48 teas. Quick, you two minus 0.5 t your team. This is the intro in each worried. So when you see inter goes like this way, you have a um Yeah, well, no meal like t squeeze a polynomial Lenny of exponential like this. Um, most of the time. Not always, but most of time. Mrs. You need to introduce a substitution. My part to integrate parts. So a woman called the UV institution. So doing this, you can introduce. I'll use different colors. So introduce Let's let you let, uh you So I'm gonna let you used to be the whole normal. So you it was cheese squared and then anything else that's left over with This is exponential time of the tea. I'm gonna call that to Devi. So Devi, is it closed to e to the minus 0.5 t t. So basically what you need to do is you need two different yet you. So you say Do you was better. Tea will be. You didn't share this part. You get to t g t and then for Mrs Dickey a tree and then you integrate Devi. So if you and integrated baby you get very that mean you interviewed his part of expression, We will give you e to the minus zero point. You know, five t delighted by the constant of multiplying. T said Rabbi minus zero point, you know, five this is gonna give you minus 20. So we need to grab my mind is you know you get minus 20 e to the minus 0.5 t. So that's the, um, Theo, That's the U substitution of. And then so for simplicity for simplicity I'm gonna defined is integral to be I call it because writing there's old thing every time is time consuming. So that means I So that means I is itcause to one over 15 676 open a year bracket to on this integral here. When you used the integration, what part is gonna be U V? This is by definition, you any time is the interview Little VT You and uh And of course, So this is gonna be one over 15 6 76 now you ve so you most about U and V. So this is you right here. Then this is the right here. So you is t squared and envy is minus 20. Time is exponential e mails to minus zero point. You're a lefty. So there you got buying is 20 t squared you to the minus. The point, you know, five t my eyes. And then this integral here will be the interval from 0 to 48. So the Samuel um so V is this guy again. So that's what it will be minus 20 e to the minus you. No point. You know, five t and then at what is what it do. You. So do you. Is here is to Todt to t beauty. Of course, you have to close a big racket here. So? So, no, this interview is getting big. Big. So once we clean this, have we obtain won over 15 676 And then, um, of course, here we have minus 20. I'm not gonna worry about limits. Like evaluating this function. It limits because, um, you to the minus 0.5 So you can see in his expression here, there's a two and in this minus 20. So you get 40 there, that's 40. And then with that minus sign outside interval, you get plus 40 and enter. Go from zero 48 and the T E to the minus zero point. You know, five t t and then it pulls out, puts clothes. Um, that interval, that bracket. Now I'm gonna go to the next page. So when I go to the next page, I'm just gonna focus, and he's part of the integral. So let's call it I Hey, Okay, let's call it I want because we already have. I was there. So this is I won. So in here we have. So we had I So we have t minus uniforms, you know, five t t. Now we use you to substitution again. So you have you is a ghost team on V. I mean, TV will be the exponential work. So this is you now You different, Should you? You get to you. Is it close to where the current CTU gets Get one duty or get sticky? And then you integrate Devi. You get V is a close to minus 20. You two minus uniforms. You know, five teeth and then I gang is gonna be u V minus the inter go feed the U Which again, uh, this is gonna be so We've already seen how to do the UV substation. So you is t on, then VSD stem here. So you get minus 20. T needs minus 0.5 t and then mine is the Inter go. So the intercourse Vidi vi, do you? This is the again. So they'll be minus 20. Going to erase this. So So That's, uh, um, so minus 24 for V each of the miners. So that's that's that's V. And then the years just gt so one. If you simplify this so you get minus 20 then you have a t the exponential. Uh, I know these. Ah, these a minus sign here. This one signed those air gonna cancel. So you get plus so plus, uh, plus 20 on an inter go. Yay to the minus 0.5 t teeth. Now, this integral here is easy to do, so you can just do that right away. So you got minus 20. See you to the minus. You know, five, uh, plus 20. Now, it isn't into go here. This'll one when integrate is you get minus 20 e to the minus. Okay, so now we have everything. And then at the end, you know, like, we're gonna evaluate its function from 0 to 48. But I want I just want to get the general idea that Theo way valued, You know, everywhere you go, messing, it's not enough space to write out everything. So we're going back. We see that. So if I drink this. So I remember these was the expression. So he sees our expression. I wanted that is the new integral. We just didn't get a page. So that means we're gonna carry over one of a 15 676 and the inter bracket minus 20. You serious? 05 t plus 40 and then 40 would multiply the new integral. So now, um, when we list, let me hoping you knew Paige. So when I open a new page So isa cause too remember, there was a constant, which is one of 15 676 Yeah, the plan is not It's morning. Very well. What? So? So so 676 year. We had a huge bracket. And then the first time here was minus 20 times t squared e to the minus zero point. You don't fly, t um, plus 40 and then 40 was multiplying, so regarded to the next to the previous page. So this time he will give me minus 400 argue minus 400 like 20 timeline eternity with minus 400 we have minus 20 here. So basically what we have is the first time would remind 20 tea time exponential minus 20. So let me update that. So here would be minus 20 t time with the exponential years miners, you know, point beautiful five t um, minus 400. So the last time here if we go back. So the last time you can see is minus two minus 400. Time is exponential. So we get e to the minus, you know, 0.5 t that causes that bracket. And I caused the overall bracket, and then I will We're actually just evaluate its function. Um, way valid from 0 to 48. Now, we're gonna save this this expression here because you're gonna need it later on. So this is equals two. I'm gonna try to group things that, uh it would be easy to your values. So 676 That's the constant. And then days you can see these exponential everywhere, exponentially. I'm gonna fact that out outside. Um, so I'll leave the 20 tea in here 20 square story, and then the 40 time is 20. That's gonna give you minus 800 times T. And then the last time would be 40 times 400. So that would give me minus 16,000. Yeah, that's right. So, in fact, if you want, you can take out the Maya sign here so that everything you become a plus plus plus. And then I'm gonna put a small bracket here so that the whole thing is multiplying. Exponential minus you. No point. You know, T and then a overall giant back. So and then I'm involving this from 0 to 48. So once you plug in this body now, it's just arithmetic. So you just plug in t equals 48. You get a number on, then t oppose you. Good number one. He's tracking us, too. So you should get you should get something like this is gonna be equivalent to 0.43 92 or rashly, um, this is rather gonna be 43 for 92%. And that's that's the answer we're looking for. So this is, uh this is the numerical value probability they were looking for now, for part B. Now I'm gonna do let me zoom out for better go zoom back in. So in here. So now for Harvey, the question was trying to answer Here is we need to determine the probable that infected infected patient will not display symptoms until after 36 cells. So the keyword used until so in terms of probability, this is gonna be P A. T is greater than 36. Okay, so p of tea is great and 26 So, um P o t bird and 36 uses the same us the inch ago from 36 Remember? Actually, I think I made a mistake. So in the first page here, if I go back in the very first stage, so I can t hear the way I define t this is not true. So t I hope I have you guys grab you kindly. So cheese between 1 50 and zero. Okay, so So this part here, So a t he's between zero and 1 50 Hey, so make note of that cracking. So if I go back here, So we discovered that, um, So, um Oliver, So integration are going from 36 to 1 50 and then you interviewed herbal f f t d t. Now. Once you do this, I'm not gonna do this interview again. So you should get this expression here. These same expression here Hey, you. Same expression. You should get that. So what's gonna be different this time? The only difference that you're gonna make here is the limits of integration. Indies expression year. Our limits. I went from 0 to 48. This time the limits are gonna go from 36 to 1 50 So? So that means you know, you'll get so these expressions gonna be in closed too. Ah, minus one over 15 six, 76 And then the first time will be a 20 t squid. Last 800 t 16,000 uh, times, times x, the exponential you to minus, you know, 0.5 t and then you about with this. So you're badly at this from 36 1 50 So that's the one difference. Why? So when you solving this type of problems, try to reuse some, subject it so you don't really than will. So and this is a really good example. Demonstrated is gonna see you at a time when you out you exam. So in here, we a little got. So once you're involved with this, you should get something close to this is gonna be roughly 0.72 50 or this is gonna be wrapping 72 0.50%. And that's what we're looking for. Hey!