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(a) Suppose the lifetime $X$ of a component, when measured in hours, has a gamma distribution with parameters $\alpha$ and $\beta .$ Let $Y=$ lifetime measured in minutes. Derive the pdf of $Y .$(b) If $X$ has a gamma distribution with parameters $\alpha$ and $\beta,$ what is the probability distribution of $Y=c X ?$

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

Temple University

Cairn University

University of St. Thomas

Boston College

Lectures

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Suppose the distribution o…

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Find the indicated probabi…

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Two components of a comput…

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Let $X_{1}, \ldots, X_{n}$…

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If $X$ is distributed as $…

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Suppose the lifetime of an…

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Let $X$ and $Y$ be indepen…

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Suppose that the PDF for t…

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Find (a) the mean of the d…

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Let $X$ denote the time to…

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Consider a random sample $…

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Let $X$ be the random vari…

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Let $f(x)$ be the uniform …

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the question given here is ah about a function with, uh they've actually told us true. Consider the ah Gamma distribution parameters. I'll find leader. And they have said that Why's the lifetime measured in minutes? So in this case, to start with the simplification, I'm going toe force trained on that. Since, uh, we know that one r is actually 60 minutes, so I can therefore write down by as a function of X. This is equal to 60 X and, uh, now, huh? You know that Why is going to be less than or equal to some value small vial that we're trying to find? So if we substitute capital while 60 ex so I can therefore right 60 X is less than equal to why. So it actually implies that X is less than equal so small. Why a born 60? Not therefore. I can say that the CD if is given by because we're supposed to find the CDF for this expression So we can take this as f off Capitol. Why? For small Why? And this will be given by a probability off. Why is less than people too small by so bryce? Absolutely. Value off capital by here. This is capital Exes Lesson. He will do a small buy upon 16 now which you can also buy the definition of CDF the prison this says f off Capitol eggs on the perimeter that we have our why Upon 60 Beata Alfa on this We're taking the representation for Obama distribution as given in the question. So now we have expression with us. Well, we have f off. Why? For small rise equal to this expression. So now I want to differentiate onboard signs with respect away. So applying that ever does we can send differentiate with respect to why on both sides that is that the religious as well as are the charges. So the left inside This will change to the off being Why off if off why into a small vial and why sure on the right hand side on gonna return this part as D by the way off the question that we haven't the right inside Which is why by 60 beetle with the parameter alpha No, the left inside. We can ride this. We know that if you differentiate the CD if you get the media four foot so this is the FBI off. Why? We're not equal what you are going off. Like Jane ruled off derivatives, some will introduce a new variable. You in such a way that I can write this as d by you off. This is if ex off. Why upon 60 be time taking as you loses with parameter alpha. And, uh, since I'm different treating this with you. So I'm gonna go over here. What? Di bi di bi off the value off you that we have, Which is saying via bon 60 beata So I hear have considered us by upon 60 meter. So then we again doing the same concept CDF uh, Dana Madoff CD of combination SPF. So Sylvie f x off you with the parameter alpha and hear the dead. If it off, I will be one. So we have one upon 60 bigger now moving for the We also know that pdf or for gonna function by the definition is given by F off X with the parameters so far here here for this particular part of living beat as one. So this expression will be one upon graham off alpha in tow. X rays toe over minus one and, uh, years to minus x. So in this expression, if I use this formula to substituted for here, then I will get a new expression for F i o fi. So if I write a part here, I really get This is equal to one upon gum off Alpha were mainly tat only and in place off X we have by upon 16 Niida So this is raised to minus one and I have easiest to minus by upon 60 Beata and into this is one upon 60 be done the constant park. So if I simplify this expression now, we can really like the whole expression in a different form, I guess by by rearranging the domes So now I want to read into says Yeah why are you stored for minus one in two years to minus y upon 16 Niida divided by Come off alpha and into this is 60 beater today has to offer because you simply find the power's off by and 60 meter. So this actually now gives me f off. Why would the better? It is not a far and 60 You go. So yes, we have found the pdf for this particular function. So we can therefore say that PDF or fly has it gone? My distribution? Yeah. Mitt Romney does. And foreign 60 meter. We're going to the B park. You know that Vaezi or two c x. So I'm gonna repeat the whole procedure for the Pardon me, and we can Therefore, I have the finance off a barn. Me as a fly off. Why? Which is the pdf me equal do if off why with tomatoes fo and civica. Because you would repeat the same steps as in part A. So this is done for the pardon me?

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