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The article "Statistical Behavior Modeling for Driver-Adaptive Precrash Systems" $(I E E E$ Trans. on Intelligent Transp. Systems, $2013 : 1-9$ ) proposed the following distribution for modeling the behavior of what the authors called "the criticality level of a situation" $X .$$$f\left(x ; \lambda_{1}, \lambda_{2}, p\right)=\left\{\begin{array}{cc}{p \lambda_{1} e^{-\lambda_{1} x}+(1-p) \lambda_{2} e^{-\lambda_{2} x}} & {x \geq 0} \\ {0} & {\text { otherwise }}\end{array}\right.$$This is often called the hyperexponential or mixed exponential distribution.(a) What is the cdf $F\left(x ; \lambda_{1}, \lambda_{2}, p\right) ?$(b) If $p=.5, \lambda_{1}=40, \lambda_{2}=200($ values of the $\lambda$ s suggested in the cited article $),$ calculate $P(X>.01) .$(c) If $X$ has $f\left(x ; \lambda_{1}, \lambda_{2}, p\right)$ as its pdf, what is $E(X) ?$(d) Using the fact that $E\left(X^{2}\right)=2 / \lambda^{2}$ when $X$ has an exponential distribution with parameter $\lambda,$ compute $E\left(X^{2}\right)$ when $X$ has pdf $f\left(x ; \lambda_{1}, \lambda_{2}, p\right) .$ Then compute $\operatorname{Var}(X) .$(e) The coefficient of variation of a random variable (or distribution) is $\mathrm{CV}=\sigma / \mu .$ What is the $\mathrm{CV}$ for an exponential rv? What can you say about the value of $\mathrm{CV}$ when $X$ has a hyperexponential distribution?(f) What is the CV for an Erlang distribution with parameters $\lambda$ and $n$ as defined in Sect. 3.4$?$ [Note: In applied work, the sample $\mathrm{CV}$ is used to decide which of the three distributions might be appropriate.(g) For the parameter values given in (b), calculate the probability that $X$ is within one standard deviation of its mean value. Does this probability depend upon the values of the $\lambda$ s (it does not depend on $\lambda$ when $X$ has an exponential distribution)?

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

Idaho State University

Lectures

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Let $X$ be a random variab…

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All right, now in the question here they're given is an expression for pdf off hyper exponential distribution. So the first, but just about is asking us to find the CDF for this particular pdf someone to write down the expression. We know that silly f and B Okay, we have to write this in terms of the variable. So he had the label is Lambda one. Then we have to. And then the fourth variable is be so well today this as, ah, integration off the given. Pdf from, say, zero West. So I want to write X as why we're here for simplification and again the valuables or the parameters as Otis and we're in debating this with Why So if I heard the expression that we have been given in the question don't buy or healed. So this is integral zeroed wax and what was substituted The values we have be number one. Eat a stew minus lander one by plus the next borders one minus B into London too. It is to minus London to buy mint. Anyway, So now we will deliver this with respect. Oh, why soapy into Lambda One is a constant. I'm digging it outside. Insulation off, idiots to mine Islam that one by upon this is mine. Islam. No one says from zeroed works, so we can cancel on the one here. Okay. And further, this is one minus B into this Islam dental. And the indication of this will be eating is to minus Lambda Tau by On mine is Lambda Tau. Okay. And the limiters, You don't ex, so I can cancel Lamberto also. So the value that I'm finally getting here is so this is minus B in the black. It now people, top element. We have meetings to mine. Islam, double necks minus Easiest was always One plus is one minus B on the bracket. We have Edie's toe minus Lambda Tau X minus one, because it's easiest to zero and this whole thing upon minus one. So if we give the minus science inside the bracket, the final time that I'm getting here now is be in the racket. One minus. It is to mine. Islam. The one X and this will be a plus one minus b into this minus. I'm giving the new military. So you have one minus. It is to minus Lambda Tau X. So this is the CDF for that? I have fun for this fabulous distribution. So therefore, toe rather finance that I can write down the CD of as this is a B is equal. We convert downtown's off limits and they're going to be zero for, say, excess lesson zero. And this will be be into bracket one minus. Used to mine is Lambda one X plus. We have one minus b into the bracket, one minus used to minus Lambda Tau X. This is going to be for X rated than or equal to zero. So we're done with the part, eh? Not moving on to the part B The question tails me. Oh, that I need to, uh, use the values. Given the question be undone. Lambda Tau Tau actually find on the castle. The value of probability of X is greater than 00.1 So true. Start with upon me. I wonder I don't the values that they have given to us. So we have been known that he is, you know, 0.5. Then we know that Landau one is 40 then. I also know that land it was 200. This is given it wasn't a question. So he wants to find probability Off X is greater than you don't mind? Yeah. The question see, is it is zero point 01 So this probably part I can write as one minus B off. This is X is less than equal to 0.1 So this case this will be now equals one minus. I can learn this as capital left off zero point 01 and finally this candidate and has one minus for finding this value, I can use the CD of dust that we've just calculated above. So, using that concept, I'm actually going toe find that value now. So substituting those values along with B and Lando Lando, this will change 2.5. The black it we have one minus is too minus 40 into this is 0.1 Get is the first part, and then I have bless. One minus 10.5 will give me 0.5 again. And, uh and I'm not gonna have one minus e stool minus 202. This is 0.1 So now if he has lived this entire expression, we're going to get the final Answer this for me. We have one minus. This is, uh, wind. Five and in the bracket were using the calculator here. So opening a black it again. And this is one minus. Yeah, we have e race, too. This is 14 2 points. You no one is going to give me going for some minus point for And, uh then I have one Does This is minus 0.5 on the bracket. The they have one more dumb so that in the record, we have one minus. This is Edie. Stool. My stool. Yeah. So we're getting the Internet's 0.40 to eight. So what do I don't find an answer Here is blind 40 go eight. So this is what we're going to Probably the venue for the second part now moving on to the tall part. The question for the tour part, see, is that, uh if X has ah, the given if as a speedy of what does the EOE effects. So we're supposed to find the expectation for the given, Pdf, You know, the affectation, the value that we have weaken Diogo. Oh, say let x one and x two Okay, B Oh, exponential landen valuables. But though, but I mean, does as London one and learned up to. So in this case, therefore, I can express X in terms off be into X one class. This is one minus being brew x two. So applying expectation to this expression and then therefore say that your ex is actually equal. We apply. Expectations only functions off. So he's a constant women like that only we will apply expectation to x one Bless again. One minus people. Um, in the saddle. Me and we have life expectations. Rex too. No for this distribution, the expectation for X one will begin by one upon Lambda one. And expectation for extra is even by one upon them. That Sofus observe these values over. Sure, I can therefore find out the main expression for expectation. So it's going to be off on land. Are one plus one minus B a born than that. So this is the answer for the expectation off this particular function. Now moving on to the next part Barbie. The question is that we're supposed to find out now the expectation of X squared and, uh, they have given us an expression that expectation off X squared is drew upon land a square. So, using this, I'm going to find out why. Continues distribution. What is the expectation of X squared? So I can therefore say that now, by definition, for continuous distribution, expectation off X squared can build on it as integral 02 infinity. It's actually minus infinity plus infinity. But we're leaving zero to infinity because the question is existing for positive function. So this is X square into FX the X Okay, so then that is equal to integral d o to infinity. This is X squared in tow. FX is the pdf, which is given. It wasn't a question. So I'm going to ride that years of spiel under one Andi eat is to minus No, no one X Okay. And then Plus we have one minus b. The lambda to it is to minus lambda Do X with the ex so we'll hear the lander one and one minus freedom that were Constance and between X Square. And here's to Tom. We're actually supposed to apply the concept of expectation of X square, so I'm gonna take the lander one outside, and this is in Diggle zero to infinity X square. It is to mine Islam. The one ex, the ex less. The next part of what I have is one minus B into Lambda to immigrant, zero to infinity X squared and Dewey's to mine is lama to expect the ex. So now this part, I can actually write as, uh the e off x squared for X one. And this becomes for extra. So I'm replacing the terms with B. Now. No one has situs. And this I'm representing as expectation off. This is excellent square. Okay. Plus, the next is one minus B in tow. Tow. And, uh, you can actually read at this whole function with removing the dorm off lambda, um, one in love that you're because you want to give that as a part off expectation of excellence, Quinn and expectation Extra square. Also, that this actually will now change to pea off expectation off. Excellent square directly because, uh, well, I'm not gonna Lamberto apart off accidentally functions, and then we have Plus next, just one minus p into this is expectation. Off extra square. So this is what we've got as expectation off x square. But it has been given that this morning we have told you of X squared is true on land a square so I can substitute or use this concept to substitute the values here. So I can therefore write that now you have X squared will be actually quitting parameter for Explain Islam, dominant barometer for extra Lambda too. So this will change to be in tow upon Landau one square. Plus, this is one minus B into do upon lambda do square. So if I read it, this is the proper form. The expression that I'm now getting is to be upon Lando and square. Plus, this is twice of one minus B one time that'll square. So this is what we're going. The expression for you off x square, but moving further. You're also supposed to find the expression for aliens off X. So we know that the formula for readings off X is given by expectation off X square. Minus the mean, which is you affects the whole square. So we performed. Boulevard is already so I'm going to write down. The value is here. This is to be by mandolin Square blessed rice off on minus B into them upon Lombardo Square and then you affects the whole Tequesta. This changes to minus bracket. We have be a born Landa one and this plus one minus B upon them. That too, the whole square. So this is what is the value for go billions? No, moving further. The body. The question states that we're supposed to use the explanation institution to find the core radiance here. So I'm going to use the formula off the meaning standard deviation off exponential distribution to find the cove aliens. So the cove, Aidan's candidate and as Sigma upon meal now for the exponential distribution sigma is given by one by Lambda upon mean is also given by one by land us If you simplify, this actually gives me one. So we can therefore control that the co variance for an exponential distribution is equal to one. Now, uh, therefore, I can use the part c and the party that we have proved about in this particular some toe find, Ah, the cool villians off the high board exponential distribution because we need to find it for this also. So now, uh, I know that the Corbin's Sigma upon Mu can also Britain as doesn't re underworld off. We have the evidence, which is to be a born land a one square places this twice off, one minus b a born London to square and this is minus. Extend the bracket. We have be off Orlando. One plus is one minus B a porn lambda through the whole square. And the whole thing is now divided by expected him Just be upon around the one plus, yeah, one minus p upon Lambda too. So I want to value this expression in such a way that we can try and simplify it in a better be So I'm I don't think this is a normal expression under the rule cure, so we can actually write down the whole thing again. So this is Toby by number one squid. Plus we have twice off one minus B upon this Islam dark blue square and minus the Stanmore. Right. You're as say, Nanda do be what I can I have to be. I like it like being number two. Bless. This is Lambda One into one minus B. The holding A born. You have longer one time that you and the square and said the bracket and the whole expression is again divided. This we're taking it under the hood now. So the expression if this got into the triple hole square again and I'm cross my playing it. So we have be into Lambda Group plays. This is Lando one in the one minus b. The holding A born Landa one them that, too. And the whole racket square Because we have taken this undergo group. So if you see, I can actually split this into the numerator and the denominators and the expression that I am goingto finally. Now get is this is under cough. You have? Ah. If I cross multiplying the four stones, we will get Toby in tow. Randle square. Bless. This is, uh, two into one minus. B induces lambda one square. Okay. And the whole thing is divided by lambda. One square on dhe, lambda to square and the right inside water means only minus one. And under the road itself, the school thing is again divided by the actress expression that we have is being to Lambda do less land a one in the one minus p. No one square back close here and upon says Lambda. One square and a few square. So this is the whole expressions under the hood here, so we can cancel these two domes, Okay. And, uh, finally, the remaining dome that I'm getting here and we'll read. I did. So I'm getting this. A strays off the black it I can read this a speed on that square. Plus, this is one minus B into lambda one square, the whole racket close upon. He would I have a feeling I'm going to bless. This is number one into one minus. It's me. And here it is. Whole square again. Okay? And this is minus one as it is so devalued. This enough for the way we can say that Generally a probability, or like being one minus Peter present probabilities. So, probabilities we know it always lie between the 11 So max to max value the gravity can take is one. So if I see the value of this whole expression on an average or the numerator off is actually at the stairs large as the denominator, and we can therefore said the asshole fraction of actually going to be greater than or equal to one approximately so then, In that case, I'm actually now want to simply find this and say that signal upon you can Therefore, Britain, as is greater than equal, wondered off price off expression. I'm substituting by one minus one. So therefore, I conclude that sigma pawn you, I'm going to be greater than or equal to one. So this case, we can see that the obedience foreign hyperextension distribution will be greater than or equal to what? No, you didn't get the part. We wanted the next part just the part f. It is talking about the old and distribution. So for the Orleans distribution, the formula again for cooling its remains the same majestic ma born mule. But for a Nolan's distribution, the Sigma is actually given by Rudolph in upon Lambda, divided by mule. The mean is given by N divided by Lambda Soudan. Cancel lambda shirt and be give the expression now, which is actually if you cancel the roots also so we can therefore get the core visions. Why sure as yeah, cool aliens over here is not actually got one up on food. And so we're done with all the bots here. And this is the final also

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