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Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. ...

Question

Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04.(a) What is the probability that the life of the component is less than or equal to 6000 hours?(b) What is the probability that the life is greater than 4000 hours?

Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. (a) What is the probability that the life of the component is less than or equal to 6000 hours? (b) What is the probability that the life is greater than 4000 hours?



Answers

Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. (a) What is the probability that the life of the component is less than or equal to 6000 hours? (b) What is the probability that the life is greater than 4000 hours?

So you have, uh, lung to light bulbs from Vienna. This light bulbs? Uh huh. Um, yeah. Can be Can be more with it's ah, life. Can we move with all of exponential functions? We mean, yeah, exponential. Probably the function with me 1000 hours so that the probability that the light bulb fails at a time is gonna be beat. The minus t 1000 is that is probably yesterday distribution over 1000. Yeah. So this is the probability that the failure time is lesson, so it burns out. Okay, They're in the south. So if you have a lot lamp to light bulbs like this. So what is the probability with both off the times, T one t two are less than 1000 hours, So that would be, uh, So we will multiply the toe, Don't know the probability functions, and then all those are into independent variables. So we can we eat today. My list, you one. We're 1000 times minus t 2000. Thing is the T one D two until the bounce are independent of each other with them simply CDs a single into role squared interrupt from zero off for 1000 if you didn't mind sti over 1000 divided by 1000. Okay, so that's where squared. So doing this, Uh, all that has, um, Until iterative today. Minus minus. He didn't mind. Misty. 1000 was valuable. Waited 1000 zero, which all these number would be, uh, minus in gray. Mine is one minus my monsieur. Was it to this year? You do? This year is one. So it is that. And then well, this is the Samos. This is one minus you today what we eat. That's great. So there is a problem. You don't need to light bulbs for anything that to light bulbs. No light bulbs, Phil. Um, within the first ah, 1,000,000 hours with the south, they have mean, like, $1000 under that. Probably the distribution. So that is, uh, now, assuming that we have one leg bulb. What? Bad one. And then there's another light bulb, not fails. This one fails Les in less than 2000 hours, de start, listen 1000 hours. Well, so let's call this time, See? And then this light bulb, his route place by another new level. So this libel has Let's say that this whole life has time. Why? When it gets switch. So what is the probability now that these two lead wolves? The old one? This new live bulb that got replaced, They probably get these two, Phil. That too, to fill in less less than one hour, less than 1000 hours. So for that what we have to dio Ah, conditioner? Probably. So it's gonna be They probably do that. Well, it's called this the life, The time here for the Israel, X time for anyone. Why at the time for this old one z So it is probably did that ex and why? Uh, they are both less than 1000 are worse. Well, so for this new time since why got replaced Observation. Very like this probably did. Brother X is less than 1000 and data? Why minus the time Where is that replaced? That is less than a now, er because ah 1000 because, well, uh, this one get burned sales in a time Islam 1000. So it is that probably given that, uh, see is less done. Ah, husband. So this is a joint project joint. Oh, really? And so well, you can be modeled with with three by rivals to help here. The probability for this one, Tofail, is, ah, interest from zero up to 1000. And then we have the distribution. Probably distribution is e to the minus. X divided by 1000. And they've activated by 1000. Now, um, these, uh well the right, we'll see. Also, um, bills. Uh, no. Has that distribution on then? This new level has the same the same distributions that you have to be. Bye bye. 1000 divided by 1000. So those multiplied on them, we integrate do y. But since, uh, why has a shorter time the the time, but White House to live. But to meet the condition is 1000 minus the time for C. Then you can integrate. See, data. What is there is 1000 x from zero up to 1000 line. Yeah, and since we're given this condition that he is less than 1000 we need to divide this probability by the probability that see fails within the first 1000 hours. So it it is like this. You see, that's very, very cousin. So on first thing to know is that here is X has been multiplied on. It doesn't appear in the bounce for Z on Russian. Why you can pull out? You can't pull out these. This is going to be able to the interval from Can they multiply that integral separately? No words on the that multi Klein. They seem to go from 0 1000 and then nature from zero up to 1000 minus e off. Today my c over throws you. I'm seating at minus. Why? Over 1000? Right on then. The order of integration is here first the y Listen on, then. Well, we're dividing all of these by this. Probably be by the problem. You got, uh, see Phil's giving the first 1000 hours, which is these number you didn't mind? See everybody 1000 BC on board assess We can notice here. This integral is precisely the same as that. Up to a change of variables from the variables of integration So that these two numbers, uh, they canceled the US one so that the total probably is gonna be this integral left there. So let's write, Why don't you go old So I need to go easier. We did a minus. Why real frozen. Why do you see this divided by 1000? And then that times you didn't like to see. Okay, if I buy 1000 you know that very 1000 on them. Why is integrated from zero upto 1000 minus e on then see, is integrating from 0 to 1000. So, uh, so here. What we have is, um but we can integrate these so control of, uh, this integral. So the minus today minus y over 1000. Um, variable. We're getting Sue. I'm 1000 minus e on then that needs to be multiplied after by eating. And I see by 1000 and all of that by 1000 on. Then when you think of that disease from zero of Groupon 1000 So okay, when you replace this volume, why, they're so we're gonna get minus me today. Well, 1000 and 1000. That is minus 1000. What about thousands? Minus one the minus minus. So, plus C right, 5000. They're minus. These are two years old, minus into the zero, which is, uh, one minus minus one soul minus minus one, Which turns out to you last one and So then that e c minus e about 5000. You see the bathroom? 1000 from zero. Up to 1000. So we have that on. Then re multiply this to through. We're going to get minus to the minus one. And then these two councils. Yeah, my Rizwan placebo. Then for design with the there is +11 times e today might see, you know, 1000 on all these divided by 1000. So this interval disease from zero up to 1000. Also these interval sequel toe. Yeah, when I get one of our feet little divided by 1000 uh, Dr Z and then for this one, well, they developed that would be minus. He did. They might see 1000 right? And then those to evaluate that he tells it. Okay, 10 on this number is equal. Thio. Well, if you place uh, yeah, over the Riviera, you have the minus there. My thoughts waas eso So we have that number and then when we replace that, we're gonna get minus 1000 e. And it's 1000. Uh huh. Minus E to the minus one. Okay, Sundeen minus this zero will be to minus is to them I, Monsieur, which is one. So this number is, uh, bad casuals. This is minus one ovary like watery on this is minus my once or what? One. So it would be a pistol. Total number is gonna be one minus one degree, minus one of some minus two. Over and that. Well, that is the probability. Not off the second situation over destination number two in the situation number two is that we have thes to buy bulbs. One is okay. And then one libel fails the libel, See? But liable is dead. She is bad on his. Gets thrown away. It gets good placed by another light bulb. Why? And so, given that the life of C was less than 1000 what is the probability than or than direct? Is this than 1000 grand? Also, that this new one Well, minus this time over there. One is there's $9000. Yeah. Also, it is that probably given, but, uh, see is less than so. It is this perfect. Yeah, that is equal to huh? She would be called one minus two over

Okay, so we're told that the time that it takes an I T department fixed the mobile tower is exponentially distributed, So that has an exponential probability distribution. It looks like this in the mind to say X, but we're not told a We're told that the expected time is three hours. Okay, But we need for the exponential and distribution That news also equal told one over, eh? So in that case, we could just solve for a is one third. It actually affects is it's part a f of X is one third, one third kneed in the minus one third X and in part B says, find the probability that it takes between two four hours Rex the time in hours, I guess I could've used tea. What's the matter? So integrate two for this distribution function. And when we get one third defined by negative one third soaking negative needs of the one third ex from before. So this is going to be he too The negative two thirds minus e to the negative four thirds

Okay, so we have a random variable. That kid's thie expected life of a computer card Get in by this probability distribution function where tears and ours. So whenever seventy nine t round one half to thirty, two hundred part A, we just want to find the I have a feeling So the most important thing is just to set up the integral. So it's just one half thirty two hundred of tediously rate with t seventy nine hundred t kitty, and what we get is ten eighty. I'm going to risk using a computer out of the system. You can use a graphing calculator whenever you want. Two for being me want Ah, the standard deviation. So first we'll find the parents and the variance is going to be given by this integral. This time we wait with t squared. I mean, I never t minus mu squared When you find this and we just take the square root and then gives us nine fifty two point four five two. The next thing we want to do is find the probability that tea is less than the average minus point one two five times the standard deviation case is just a number. So we'll just set a Senate girls just one half, too. Well, whatever this number is of artist distribution function. And what did we get? We get about zero point five for Teo and then finally your dean, we want Teo find the median. So we find the value in it since that this integral equals one half. And when we saw for M again, she's to computer out of her system. Or you could use a graphing calculator. You get M is equal to six five, six, one over eight, which is eight. Twenty point one two five.

Okay, So he told the hours of the wife of the device before it dies, is exponentially distributed to follow some sort of exponential distribution. T is in zero ten. Thirty. They were told that the median number of hours before this divide its dies is five. So that's in for media and five hours. Sorry. Four hours singing this for hours for you. Okay. So again, we want to find this value of a before we can actually do anything with dysfunction. So let's set up our equation for finding the median. T is one half, but here we actually know and is five, four. Yeah, What? I'm thinking fire is four hours. Okay, so we can evaluate this. This is negative. I need to The minus eighty from zero to four is one half too soft for a substance. One minus. Even the marcus forays one has Do you have so hee to the four days to? So a is one forest log, too, Which is about clear point one seven three things. Great. Now we want the probability that the device last longer than five hours. Probably tease greater and equal to five. Well, this is just the inner girl from five to infinity. Ah, here a point one seven thirty three e to the minus your point one seven three three e t. And I'm just going to throw this in my calculator and we get Syria point for two. It's a probability that'LL Last song is by ours.


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