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A mode of a continuous distribution is a value $x^{*}$ that maximizes $f(x)$ .(a) What is the mode of a normal distribution with parameters $\mu$ and $\sigma ?$(b) Does the uniform distribution with parameters $A$ and $B$ have a single mode? Why or why not?(c) What is the mode of an exponential distribution with parameter $\lambda ?$ (Draw a picture.)(d) If $X$ has a gamma distribution with parameters $\alpha$ and $\beta,$ and $\alpha>1$ , determine the mode. $[$ Hint: $\ln [f(x)]$ will be maximized if and only if $f(x)$ is, and it may be simpler to take the derivative of $\ln [f(x)] . ]$

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

University of North Carolina at Chapel Hill

Piedmont College

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Lectures

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A mode of the distribution…

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The mode of a continuous r…

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The mode of a discrete ran…

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Let $X$ have a gamma distr…

Now they have said in the question that more of continues distribution is evaluate star that maximizes f off X. So the first brothers are what is the mood off normal distribution. But parameters are mu and sigma. So for the first part, they have told us tow find them war for the normal distribution. So we have normal distribution and the parameters are the mean mu and attended the standard deviation sigma. So we know that since, uh, the normal distribution okay is symmetrical because we know that, uh, the area under the curve is divided into two. Equal house says that you know the area under the robe lady on Belikov is equal to one. So the call of this began, therefore say that the center line presents I mean median and were all the three at the same time. So far, a normal distribution we can say that mean is equal to median, all right? And that in all sequel toe more so since the normal distribution meaning more than equal, we can therefore see that the more for normal distribution is actually who'll do new. So this is when the first part no moving over the body be the question states that, uh, does the uniform distribution with parameters. And we have a single more why or why not. So, uh, to simplify our wants at the spot, we're going to focus on the uniform distribution. We can say that the uniform distribution you d I'm writing here, they have, ah, constant value between the elements. Uh, say yeah, me ask the question. So that is why I began. Therefore say that all values between and B is therefore, the more off the continues distribution that, you know. But right now, more than Canada's distribution, we're talking about a uniformed distribution. So since we can see that the values off between A and B are more four continents distributions that therefore we can say that since there is no particular maximum value Oh, no. In this uniformed distribution, So we therefore conclude that, you know that is no. Yeah. You're saying that there is no war for uniforms distribution. Okay, Now, this is a villain, with the second part moving on to the third part of the Bard. See, the question states that what is the word of an exponential distribution with parameter lambda? So, um in this case now do the third, but everyone forced. Consider the pdf off exponential distribution. So the pdf off an exponential distribution is given by we have small FX that is equal toe since Landry's the parameters we have Lambda Edie's to minus lamb, Dex. Now, as for the maximum minimal Curie, we can see Oh, Morris actually, the value off export which we can have, you know FX is maximal, so we can therefore consider the fourth day inevitable. FFF X is equal to zero to find out it is maximum act which value so in this case, have to find on the derivative off this, pdf or here with respect wasted finding the dead. If you don't keep land as it is, they never do a fetus to mine Islam. Diack says he needs to mind slam dunks and into Internet of a defense minus Lando, which is equal to zero. So by this expression like and therefore say that, uh, we have minus Lambda Square, it is to mine Islam die access equal to zero. So to simplify areas to minus X is equal to zero. So we take Eleanor zero, and then we take the reciprocal. So we actually get the value off excess zero. Which means that for an exponential distribution, the more is given by zero. Well, for the ah, for the body, they've told us to find out. Uh, like they have said if excessive government distribution with parameters, I'll find Peter. That Alfa is greater than one. They prove to find a more for gamma distribution. So fine. Now start with the com institution part the need part over here. I want to start again here by writing the pdf off the normal distribution. So we can say that pdf off the gamma distribution is given by you Represent that with F X is, um, invaluable. And the parameters are I'll find Vita. So this is equal to one Upon we have B does rest well, fun into your gamma off alpha. And we have expressed well for minus one with CDs to minus X, divided by Mita. So to find the more again, we're goingto use the same concept that X will be begin friend Mort if excess maximum and that it's possible if fx and maximum. So we are again considering the derivative of X is equal to zero So if I find the date of it off this with respect wigs, the initial part of Constant, I'm gonna decide that as artists. Okay, not this angle. Differentiate by blind the you rule of the elevators. So we have ah extra stool for minus one as it is, and the devil to off the lease to minus X by be tires as it is into the internet of it off. This expression is minus one by Buta. Okay, then Plus we have it is to minus X baby die as it is and the video off extra stroll far is going to be found minus one into extras tour for minus two and this thing is equal to zero. So now if I this constant, will go in multiply 10 And if I take the expressions what we have on the other end so I can therefore dying down that we have exist well, for minus one is to minus exploit Beat upon Beata can actually be the denies equal toe used to minus X by being tough into our four minus one in tow Accessory store for my scoop. So here we condenser these storms right and again. Ah expressed role for my honest work and simplify for though I can say this is extra stool for minus one. Ah, Phone Beata This is he will do our four minus one. And this we can ride as extra stroll for minus one upon x Again This part also gets cancelled So I can finally say that I'm getting X value as Beata in tow and four minus one. So therefore we can see that more for gamma distribution is given by Peter in tow and for mine this one. So we learned with all the four parts of this particular problem.

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