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The article "The Prediction of Corrosion by Statistical Analysis of Corrosion Profiles" (Corrosion Sci., $1985 : 305-315$ s suggests the following cdf for the depth $X$ of the deepest pit in an experiment involving the exposure of carbon manganese steel to acidified seawater:$$F\left(x ; \theta_{1}, \theta_{2}\right)=e^{-e^{-\left(r \cdot \theta_{1}\right) / \theta_{2}}} \quad-\infty< x<\infty$$(This is called the largest extreme value distribution or Gumbel distribution.) The investigators proposed the values $\theta_{1}=150$ and $\theta_{2}=90 .$ Assume this to be the correct model.(a) What is the probability that the depth of the deepest pit is at most 150$?$ At most 300$?$ Between 150 and 300$?$(b) Below what value will the depth of the maximum pit be observed in 90$\%$ of all such experiments?(c) What is the density function of $X ?$(d) The density function can be shown to be unimodal (a single peak). Above what value on the measurement axis does this peak occur? (This value is the mode.)(e) It can be shown that $E(X) \approx .5772 \theta_{2}+\theta_{1} .$ What is the mean for the given values of $\theta_{1}$ and $\theta_{2},$ and how does it compare to the median and mode? Sketch the graph of the density function.

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

Temple University

University of North Carolina at Chapel Hill

Oregon State University

University of St. Thomas

Lectures

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well, the Syria for the depth of the deepest speeches given it wasn't a question here, so I'm just going to like that part. What is given to us? We have C g f f off eggs, and the parameters are Tito wanted Dr. So this is given by even is to minus eat a stew. Yeah, she died that part again. Yeah. So this park is actually even as these two minus eight days from minus, we have x minus t dewan upon you got to and, uh, where the limiter mine isn't pretty less the next. Less than infinitely. So they've also given the values for kicker on anti Toto Tito One is actually knew here, which is given as 1 50 And he that war should ever since Sigma that is given as my Andy. So, uh, starting with the first part, they told us to find out, Uh, the probability that the debt off the deepest pit is at the most 51 50 orders at most 300 orders between 1 53 100 So starting with the force part, we're here when they say at most 1 50 means therefore to find be off X. There's any Pollux 1 50 So by definition, we can see this is directly given by CBS Fracked X equal to 1 50 So if I substitute our excess 1 50 in the given park and some of the values off dominant needto, what I'm gonna get us eat is to minus 30 days to minus. This is 1 50 minus 1 50 by Landy. Okay, So if you simplify the whole park, this gives the answer as these two minus one. And the simplification is folded in desolate for mass pouring 3678 So this is the answer for that. Most want to be part next year. Poor to find the valley for a tow Most 300. So following the same process, this would be called CDF at X equal to 300. So substituting the values will get us to minus. It is to find this. This is 300 minus 1 50 defined by 90. So if you simplify, it won't get, uh, the hold on simplification in the galaxy, and the value will be going 8 to 78 for the live. Also want to find the probability that it lies 1 53 100 So we can represent like this. So with the definition off CD, if you can drive, this is F or 300 minus f off 1 50 So they found all the old devalues all yours. We condemn it, Lisa, prior to this 500.8 to 78 minus 0.3678 So the subtraction will give me the value as 0.46 So you get in with the park. You know, one of the part B. The question states that ah below what value with the depth of the maximum will be observed in 90% off all such experiments. So that means 90% the product is known to us. So this is the real divorce problem that they have given. So in this case, I can write down the definition on dhe. Some small value extent we're supposed to find is also represents, Please. Cdf off. That's more value X and the probabilities only Eunice. 90 portions of each of the comm 900.9. So if I applied if affects over here, the value that we have, you have Edie's to minus ease to minus X will remain like that. Only minus this is 1 50 upon 90 on. The wording is equal to 0.9. Survivable. This your here on the side will become Ellen. So I can therefore say minus off either is too minus X minus 1 50 by 90. Would actually be goingto ellen off wine tonight. So again, this minus. I give on the right inside, and I cannot gain. Apply one more Ellen for this ease. I can therefore see that, Ellen. So, uh, we have Edie's to minus off X minus 1 50 The 90 is equal to minus off Ellen point tonight. So finally, again. Minus off X minus 1 50 by 90 Will again change to Ellen off minus ellen off 0.9. So from this expression, if I simplify this, I'm going to get now X minus 1 50 upon 90 is actually poured over. Simplify the attitudes of the calculator. You get 2.250 So the value off X is now equal toe 90 in tow, 2.250 Bless 1 50 So the value that we're getting for X here, there's 3 52.5 so well in with the part b going on to the park. Si They've asked us to find the density function for this given CDF So we know that density function FX is given by the dead. Ever give off given CDF So if I planned an element of your for this the letter Madoff, it is totally Edie Stowe as it is. So it is to minus years to minus. We have X minus 1 50 upon 90 in the you know, there's a video for this Really? Minors off Edie's to minus X minus 1 50 by 90. Yeah, and, uh, for the internet of the Gulf destruction is going to give me minus one of our 90. So the final FX if I'd I descend on Lucky down and he got only don't use this value the Nancy will be eating is to minus is too minus This is X minus T doa wanted a do minus and minus would come plus So we are again. It is to minus X minus Tito one upon ti da doo and then the world in divide by nine days he dug. So this is the expression for the density function now moving on to the next part for the party. Um, the question is saying that, uh, the density function can be shown to be yoon immortal or which is a single peak of ah, what value on the measurement access does the speak occur? Which is, actually want to be the value off more. So we need to find out this. So I can say that to find the maximum value we need to send the dead a bit of off perfect equal to zero. So therefore, ever consider abduction fax is equal to zero. We also know that if I for fixing CODIS, then though Law of effects on the island of FX will also increase. And therefore, um, well, to consider Ellen off if affects overhears. Okay. And I'm gonna find out the dead Haven't you off this part? All right. And I'm gonna take this equal to zero, because instead of taking a forfeits, his battery take and enough f or fax it to be easy for us to simplify the whole expression. So now this part. If I elaborate this we're here. This is D by the ex off. We have Ellen off FX of assures that we have, uh, with these values over here is one by I'm d We have Edie's to minus CDs to minus X minus 1 50 by 90. And this is again into Edie's to minors Thing X minus 1 50 by 90. Okay, and this thing is equal to zero. So this part, I'm going to love it now on the next page. So if I apply the Ellen property here, you know that Ellen off ABC can organize Ellen off a plus, Ellen off B plus Ellen off. See? So if I you d bindi eggs as it is, and what was planned is you have Ellen off one by 90. Okay, then I have plus Ellen off eaters to minus CDs to minus X minus 1 50 by 90. Okay. And then I have Father. Plus, this is Ellen off Edith's to minus X minus 1 50 by 90. And holding is zero. So, uh, now when you take the derivatives, But before that again, there's almost African. He was supplied the Ellen property and enough idiots. Do we know the Baba will come down and Ellen off Easy called one. So the expression will now changed well enough one by 90. Then this will be minus eat a stew minus off X minus 1 50 by 90 and is only minus X minus 1 50 by 90. Holding is equal to you know of our planet. Innovative, Ellen, off on my 90 is constant zero minus eaters to the little bit of this scene. Why didn't we know? They're a bit will give me minus one by 90 and we're here. This will also give me minus one by 19. So this is equal to zero. So if you simply find the whole expression, what are you going to evaluate? Your is the 91 water the Titans? I. So what finally means you're is Edie's to minus X minus 1 50 by 90 minus one is worth zero. So therefore I have Edie's to minus off X minus 1 50 by 90 is actually one. So we can therefore say that minus off X minus 1 51 Langley is heading off one. But you know that enough one is zero. So I can therefore say that minus x last 1 50 by 90 is equal to zero. So I can therefore see that Xs equal do 1 50 So this is a value at which we will consider the more because you caught the value of excess, the positive and the maximum value showed now moving on to the next part For the part e the question states bag, we're supposed to, uh, like you've been given the expression for the mean for this particular distribution. You Vegas expression is given to us. And afterward, then what does that mean for the given value of the dollar ante data? So we need to substitute devalues over there, and we need to compare it with the media and more and also sketch the graph for the density function that were found. So you start meant that the expression for expectation what is given is ah, your fax is equal to 0.577 to Peter Do plus Steve one. So if I substitute the values of teacher $20.1 here. So this is 10.5772 in tow 90 plus, this is 1 50 So the value will get Here is 21495 This is the exact valley off me back down and Tito now for the median Uh, we're going to consider that the media is going to lie because it's the center. So the median will be exactly a point fire. So we said that the CDF will be exactly a point fight. So we need to find the value of X where this is B at 0.5. So again, uh, using the concept ever fixed, we have it is to find those keys to minus X minus 1 50 by 90 is why and fight. So simplifying this in a similar way. We have minus CDs to minus X minus 1 50 by 90 is given by Ellen off 0.5 and found our e days to minus X minus 1 50 by 90 to be minus eleanor 0.5. So, Father, I can say that therefore minus off X minus 1 50 upon 90. This sequel do Ellen off minus at an off white fight. So I can therefore see that X minus 1 50 upon 90. If I simplify this using the Cal Tate Oh, we will have wind 3665 So therefore, X is now you will do 90 in tow. Born 3665 And plus this is 1 50 So I'm getting the value of excess 1 82.99 So this is the medium for this particular problem. Now, the more we talk about, the more that a girl's at 1 50 So we're supposed to find f off Excite 1 50 So for the more you know, the maximum value that one for me. So therefore, don't go find F or fix at 1 50 So this is 1 50 with 1 50 end Landy. So your subsidiary doesn't the expression we have eaters to minor self. It is to minus zero in two years to mine a zero on the whole thing, divided by 19. So this party valued in the cath little you get 0.409 So, uh, this is the more for this particular problem now. Next is therefore to sketch the graph for given Pdf So for the sketching of the gravity to find out the ski. So for the scale, we already the y axis because the Y axis values you just found that at 1 50 to 0.409 So it is goingto range from sincere or 2.5 Okay. And for the x axis, the scale your take from forced to the 99% die. So no, I'm not your baby from force to the 99% day because you were talking about normal distribution. So we're going to find out now a four fix, but the first words and I'll, which would be like 0.1 and, uh, if affects with 99% dance, which is like wind my nine or sure. So using the same concept we simplify. We have easiest to minus 30 days to minus X minus 1 50 by 90. This is 0.1 on dhe, least minus used to minus X minus 1 50 by 90. This will be a 0.99 So you've simplified these expressions are live 4 to 5 times. So what begins Wednesday? Libby can consider the value here as X minus 1 50 by 90. We will actually be equal to minus off Ellen off. Minors off Ellen off 0.1 And similarly sure B R x minus 1 50 by 90. This will be equal to minus off. Ellen off minus off. Ellen off, joined by nine. So therefore, from here I have X is equal. Do. This will be 90 in tow. This if you valued in the calculator says minus 1.5271 bless 1 50 So this value of X is 12.561 And here X would be equal to 90 in tow. Bracket. This value of you value it is in the Kelsey just 4.6. Bless, this is 1 50 So I'm getting X over here as 5 64 So if you see the ranges from 12.55 64 so I will consider the range off X axis safe from zero to approximately 600. This with a range. So now I'm going to take the high ground. The half of what you have already made it. So So I'm just going to the size the graph and be stood here. Yeah, So if you see the stuff well, the XX is over here. We've taken the names from 0 to 600 the rainforest for fixes from 0.5 and now for a substitute of the values off X in ffx, then getting their different range on. You can see that this shows are normal. Execution coaches area under immunity. So this is what you brought up? The graph for this particular pdf

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