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The time, X to failure of machine has Weibull distribution with pdff() = (4) (#)" Compute the probability that the machine is still functional after time 5 giv...

Question

The time, X to failure of machine has Weibull distribution with pdff() = (4) (#)" Compute the probability that the machine is still functional after time 5 given that it is functional after time 4.Hint: First; identify the shape and scale parameters from the pdf:Nonc of thcsc0.77880.8859

The time, X to failure of machine has Weibull distribution with pdf f() = (4) (#)" Compute the probability that the machine is still functional after time 5 given that it is functional after time 4. Hint: First; identify the shape and scale parameters from the pdf: Nonc of thcsc 0.7788 0.8859



Answers

Let $X$ denote the time to failure (in years) of a hydraulic component. Suppose the pdf of $X$ is $f(x)$ $=32 /(x+4)^{3}$ for $x>0$
(a) Verify that $f(x)$ is a legitimate pdf.
(b) Determine the cdf.
(c) Use the result of part (b) to calculate the probability that time to failure is between 2 and 5 years.
(d) What is the expected time to failure?
(e) If the component has a salvage value equal to 100$/(4+x)$ when its time to failure is $x,$ what is the expected salvage value?

Yeah. So the expectation of ads vehicles Enjoy a girl from 0 to 4 X. Times the probability density function. And this is just that's cute. Times Tax -4 Squared DX. Krygiakos, 15 over 512 NJ. Girl from zero rodeo floor. That's extra faith miners. Eight extra fourth plus 16 Excuse Dprk's When she goes 15 over 512 Characters to 6/6 Plus 8/5 extra 5th class four Arabs to force and from 0 to 4 enhance their Sequels to The expectation of x squared equals integral from 0 to 4 X squared times the probability density funds. And this is just X. To a force Enhance This. Eagles 15 Over 512 and enjoy a girl X 26 miners, eight X 2/5 plus 16 X to the fourth Dx And the sea coast. 15 over 512 Acts, over seven Miners for over 3, 6 Plus 16/5 Extra Faith. From 4 0- four. The answer is 32/7

One of the main things about a probability density function is that the area under the curve must sum to one over its entire domain. And if we integrate 1/5 with respect to X, that would give us 1/5 X from 0 to 4 and substituting in. Now we'll end up getting four fits minus zero. And so, of course, that would just leave us with forfeit. And because this does not some toe one, then this cannot be a probability function because it needs toe contain 100% of the likely outcomes, and this one Onley contains 80%.

Determine if this is the probability density function here. We first want to look to see that it is positive, which it is. It's continuous, which it is. So then the last thing we want to check is that it sums to er the area under the curve is equal toe one. And so let's anti derived this so we could split it up is 4/8 and the anti derivative of that would just be one half X and then minus X over eight. But that would bump up two x squared over 16, and then we're integrating from 0 to 4. So plugging four and first, this will give us half of four, which is to minus, um four squared 16/16, minus one. And then, of course, if we plugged in the zero, that would be a minus zero there. So all of this does some or combined toe one. So therefore the area under the curve here is equal to one, and this makes this probability density function

All right, We're looking for the probability that the lifetime of machine is less than six were, given that we have an open intervals 0 to 40 for the lifetime of the machine. And that half of axe is equal to quantity. 10 plus acts to the negative second because it said that it was proportional. That means we're gonna have a constant. So we'll call it kay out in front. That's the proportional part we're gonna need to solve for K to figure out what that is. Um, And then we will need to figure out again. Six being the key number therapy in less than six. So to start with, we're gonna integrate, um, from 0 to 40 to find the overall, the whole off the machine, which is gonna then be 100% or equal toe one. So we're gonna go from 0 to 40 of our function, and that needs to be equal toe want. And again, we'll put our constant here outside for now. Ah, so we're gonna take the anti derivative over 10 plus x the negative second. So it'll be negative. 10 plus acts to the negative first, and our constant K is still there. Um, and we didn't need to plug in zero in 40 and again. Still gonna be ableto one in the end so we can solve. Okay, um, this is going to give us a negative one over 50 minus and negative one over 10 and that still gets multiplied by K. Also equal to one. Um, rearranging this because we have two negatives will end up with 1/10 minus 1/50 with Kay equals toe one. Um, so bye. Getting a common denominator here, we would end up with five over 50 minus one over 50. So that's going to be for over 50 okay? Equals one. And then, by playing by the reciprocal on both sides, we find that our value for Kay is 12.5. So that's gonna help us later on, because we're gonna need to know when. Use that kay. Now we're looking for the probability that it's going to be less than six, so we're gonna do essentially the same thing. Except now, um, we're gonna go just to six as opposed to the whole 40. Okay? We still have our kay and our function this time. No, we're not set any equal to 100%. We're trying to figure out the probability that it will go from 0 to 6. So it's not equal to anything. Um, this is going to give us same anti derivative, so we'll end up so with Kay times the negative 10 plus x two. Negative First. In this time, though, we're gonna plug in zero and six. So when you do that, we'll end up with, um negative one over 16 minus negative. 1/10. I still have the K. Well, plug that in the end. Um, and so now again, flipping it around cause we end up making those positive. We have 1/10 minus 1/16 and then we're gonna multiply by K. At this point, we can go ahead and just say what K is. It's gonna be the 12.5 and the 1/10 minus 1/16. They will share a common denominator, Katie. So that means well underway. With 80 or eight over 80 over here, eight over 80 minus five over 80 multiplied by our 12.5. And when we do that, we end up with approximately point or seven. Which is the answer C huh?


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