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Problem 10 Medium Difficulty

(a) A type of light bulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean $ \mu = 1000 $. Use this model to find the probability that a bulb
(i) fails within the first 200 hours,
(ii) burns for more than 800 hours.
(b) What is the median lifetime of these light bulbs?

Answer

A. i) 18$\%$ ii) 45$\%$
B. 693 hours

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Video Transcript

if Okay, so for this question, we're trying to model the behavior libel flight when it is going to fail and you've given the average you every time for hope is 1000. So the average is mu is it causes 1000 hours. So that's how long like love is supposed to last four before it sails. And here we're supposed to use. So we're supposed to use the probability Dent's function as an exponential. He's defined like this. Run over me on each to minus one of them. You know, t that is, if tea is greater than or equal zero otherwise of teal zero for all t lesson zero. No. So now Roman A what is the probability that, um, a lifeboat lost $200 So to have it his question uh, it's the same as doing what is p of t. T. Is a friend unbearable? It's less than 200. And then translating this into an integral it becomes and to go from 0 to 200 and then you have so effort. One of a mule is one of 1000. So on over 1000 you to mine is one of them You know, we just mine is one of the 1000 t t. I'm only integrating 40 Greater than it was zero, because they're after is not zero on every t zero house. So now clean myself some room here. So basically, now evaluating this inter go. So the one of one thousands of constant so that will come outside. So you get one of 1000 from 0 to 200 and then you have e to the minus one of 1000. Oh, t d t So involved in decent guy Wilkes ST obtained one of 1000 and then the integral itself becomes you get a B to D minus one over 1000 t and then divided by the cool, efficient and front of tea. Let me notify minus one over 1000 and then I have to evaluate from 0 to 200. No, uh, you can clearly see that these one of 1000 days when it was cancelled, But you will obtain a minus. Sign there. So? So that mean you will get you to D minus one over 1000. This is supposed to be zero multiplied by t. So the apple and its two hundreds off 200 years. Well, there's a minus. Sign here, too. So and then plus, so minus minus. Well, come in. Plus. So you +20 when I plug in t equals zero, So I go one. So you would just get close to, um Now, some of us are gonna cancel here. So used to you know, that these two of the cancel I cancelled into the one here. I can see two here at five. So that means I will obtain minus the exponential off. Mine is one of the five. That's one we just wrapped in quarters, too. Zero point. It's mine is you know, 0.81 87 plus one. So, Bradley vehicles, too. You don't 0.8 13 or two of 18%. So if you can read that into percentage, So And, um, so these are the answers we're looking for. So that's the percentage light bulbs that are going to fail within the first 200 hours. Now, moving on to be, uh, so Roman too. Not be the Roman, too. What's the problem? Is that you know, a life of last for more than 800 hours. So this is the same us asking what is p. O t. Gray than the 100? So that's the question we try and cancer here. So which becomes these the same I'm doing interval from eight under to infinity and the ever do you hear? So that's the integral we need to worry about. So, um, open a new tab here. So now, during this interval, So we do from you're trying to answer the question. What is beauty? He's good in a So which I said earlier, this is equivalent to integrating it and ridge to infinity, one of the 1000 you to the minus one for so this intra go, um, is equivalent to change colors. So this is the outfit changes into a limit. So which will become? I always wanted one of 1000 outside so and then I'll changes into two image. So the reason why I'm doing this is because I don't like dealing with infinity. It's this one here, so gonna changes in furniture to x like this. So I'll be integrating from 800 two X and then you two minus one off TD T. And then that after I can take it. So interrogating we obtain one of 1000. Once I integrate this part, I will obtain I see off my limit here. So limited. Excellent, Fanny. Uh, so I got each to the minus one over 1000 t divided by the coefficient in front. Off TV is minus one for 1000. And then, of course, you value it is from 800 two X. Uh, doing this, you can clearly see that these 1000 and 1 1001 tough cancel. But you gained his mind. Sign here, so I'm gonna create myself some room here now. So this is a Kremlin too, So a minus. And then you got the limit Excluded infinity. Now we have a fucking T equals X. I get E to the Linus, one of 1000 X. The plug in X equals 800. I mean, tear was 800 so I get minus Yeah, to the minus one. Over 1000 times 800. And then close bracket. I pick one now, uh, so once I take the limit, these one is a constant. So that would just be itself. This part here, one of 1000 doesn't matter. So it's like it is minus infinity. So it is Will go to zero the Limited. I will go zero when ex ghost infinity and then for these one you just need to evaluate. So I'll cancel the two zeros Year two zeroes here. And then, um, you divide, um, a life by to you before and l did by 10. Way to a good try. So this whole thing will become minus and then minus digs Financial. Oh, minus four or five. Okay. So you can see that denial signs are gonna cancel giving you a positive number. Now, the ex addict potential off minus four or five. Once you've argued that you 10 0493 or roughly 45%. And so a dish is the This is the is the fracking of light bulbs that are gonna last for more than 800 hours. Now, um, moving on to be so be we need to calibrate the median. So? So whether it's meat iss What? What is means me? Expand it. I love it. So be it. We need some real number x site that when we integrate from zero to excellent one of the 1000 you didn't mind is one of 1000 team DT way Should get 1/2 or 0.5. So what kind of x value? This is the one we're looking for here. What? What? X file. You lies exactly in the middle to immediate means, you know, in the middle. You know, assuming that, you know, interrupt this linear. So what X value will honor this integral. So doing the inch ago, doing intervals. So we'll obtain, um pushed in 1000 outside. And then I was in the inter go from zero X. You get the to the minus one over 1000 1000 t and then divided by mine is one over 1000. And then you Valerie JJ from zero X. And, of course, it's your clothes to dry inside. Easier of my five again, as before, one of the 1 2000 canceled, but, you know, gaining my sign in the process. So now I'm gonna create myself some room again. So, doing this inch ago, um, you should get something like I'm gonna we do the interview. And when a plug in X equals zero, X equals exp ain t equals zero and teakwood exit. I get minus E to the minus one over 1000 time and X and then minus minus become plus the knee to the zero if you're just gonna be one. So when you plug these t value equals zero into here, you just gonna get you're gonna get one. So now, uh, let's move things around. Tow this term with a minus, I will come here, and then I'm gonna move these times the 0.5 years to decide. So it means, uh, one minus 15 will be equivalent to you to the minus one over 1000 X. So this once you've given 0.5. So you get points. Is it close to the E to the minus one over 1000 next to SOCO accent. It's a question. Um, you took national longer than so. Taking national walk. Yeah, we obtain long 015 is this was too long off this whole thing. So the right answer. I wanted alone. You're just gonna get minus one over 1000 X and then solving for X. You just, you know, get rid of this one. So I'm gonna hold supply by minus 1000 together. So multiply the whole thing. Multiply by minus 1000. Let me do it. Isn't different color so minus 1000. So minus 1000 and 1000 those are going to cancel. So the left hand side, you just get X And then mine is 1000 times long Of 0.5, you get minus 1000. Time is long. He's the X value we're looking for. So therefore actually is roughly now you You put this into your culture. You should get something that is approximately six 193 and that is a median. That's the median valuable looking for.