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Rework Exercise 32 by using Theorem 2 to determine the joint probability density of $Y_{1}=X_{1}+X_{2}$ and $Y_{2}=$ $X_{1}-X_{2}$ and then finding the marginal den...

Question

Rework Exercise 32 by using Theorem 2 to determine the joint probability density of $Y_{1}=X_{1}+X_{2}$ and $Y_{2}=$ $X_{1}-X_{2}$ and then finding the marginal density of $Y_{1}$.

Rework Exercise 32 by using Theorem 2 to determine the joint probability density of $Y_{1}=X_{1}+X_{2}$ and $Y_{2}=$ $X_{1}-X_{2}$ and then finding the marginal density of $Y_{1}$.



Answers

The joint density function for a pair of random variables $ X $ and $ Y $ is

$ f(x, y) = \left\{





\begin{array}{ll}







Cx (1 + y) & \mbox{if $ 0 \le x \le 1, 0 \le y \le 2 $}\\







0 & \mbox{otherwise}





\end{array} \right.$

(a) Find the value of the constant $ C $.
(b) Find $ P(X \le 1, Y \le 1) $.
(c) Find $ P(X + Y \le 1) $.

We are being asked to check what out this function is a valid john density function. So in order to fight and some that me to commit you to a total integration over the function would've spread both X and y. And if the answer is because one then this function is valid joined this function now. So we're going to do our double integration by the function, and I'm just going to treat this one first. So this one first, we are going to have Tio and then know that because I'm dealing with this go to X. Why's assume a constant? So we have 0.1 and then we'll speak to exert yourselves five and then e reza bars over five x minus 0.2. Why for eggs? Well, is infinity to zero. Do away. No one you simplify the about Haven's opens 10.5 and then when access infinity, this core function goes to 01 x is zero have e minus 0.2. Why do you know why now when we strike to interview over Why then we are also going to get because this one is a constant We can just pull it out and then this is going off. So we're having infinities. There are So it's one by zero point fried and then he mine is fine too. Why away? And when you interviewed to respect you why you should be able to get 0.1 divided by 0.2. So is opened. Fight really planes one divided by men is 0.2. He's man is open too. And then value's off asses. What? Infinity and zero. So when he said If I again going to get 0.1 divided by minus this time, that's what you 0.1 and then when? Why is infinity you have in zero and one ways. So the results in inside wass When certified this one, there's notice what one? So we have been able to verify that this function is did a joint density function. Now the next questions they should be able to find Why want me t off? Why one And then I always say that this one because if they don't this defection and they do have the corresponding x. This can also be written us Why with one and then s great our coaches. So what? I'm going to The unity integrates over white values from 0 to 1 And eggs varies from infinity to you also won he man, Is there a 0.5 s not 02 Why the eggs? Y so just ancient street first with the ex girls having infinity one. And then is there no point? Once when you integrate this one dissipating X, we haven't minus 0.5 e man is over five Excellent. 0.2 way And then for values ofthe earth being infinity to zero. And then why now when you try to simplify the horror of this should be good to get infinity one and then 0.1 Honest divided by 0.5 You have negative Negative sign there already. Now when ecstasies infinity This route in bizarre and I went excess. So this whole thing E man is kind to what? No one is simple Fire this whole thing again. You should be able to get a 0.1 divided by 0.5 because this and this whole cancer we have e man. Is there a point to why the wife? So when I try to interview this one Station X. We should be there, which gets opened one divided by 05 as always there. So plans. We have one divided by men. Is there a point to? And then e man Is there a point to wife for Ed's variance? Infinity to one. And then this whole thing can also be certified at 0.1 divided by man. Is there a 0.1 now when? Why is infinity we have there? But when? Why is one? The result is what 0.1 So they want 8187 So when is certified? Cool off these you're getting minus one minus Sorry. Zero money Sorrow printed 81 eats seven. So the finances with 0.187 The next question's that China Tio Tio. It's less than you got to know why less than he called for. So just integrating X and then because this is the highest volume off. Why do you have for day and so on? Defense? Xs. What's to answer? Because they're twins. One in man. Is there a point? Five packs Money's. Is there a point? Two way the eggs away. So there's going to treat when you spend two X as we usually do first. So we should be able to get infinity falls. There are. And then when you interviewed this, our respective X the wise treat that as a constant haven't 0.1 minus 0.5. The man is open to five eggs. Money 0.2 Why? And then for exuberance from to is there do you now inside to some fight, asshole of this you have four years are 0.15 Now when X is to we have in Venice, one man is opening soon. Why anyone excesses are e and then right? No, I don't want to simplify this whole thing. So this Walton can be sent by us. For us. There are 0.1 minute So five and then this whole thing up. So e minus one Because of this one can write too. Why minus one is 0.2. Why do you want on this whole again? Can also be simplified. Us What? Um, so cause there's 0.1 0.5 and then you can say in minus one on this one because I'm trying to fact that this world and that so hee minutes or point to Why? So I'm just trying to integrate this one. Expect to why this is a constant this also constant. So, Seamus, what is their opens? One of these negatives opened five e minus one on this one. And then when we integrate this one with a sports way, we haven't won a better man. Is there a point? Inman is zero. Why? For exams to size four and a zero. So kind trying to simplify the whole of this. You should be able to get zero one's wand. But this must mend. Isn't this having 0.1? And then we have in minus one, minus one and then when? X So when y is for when you write four. When y is for having e man is opens, eat. And then when y zero we have Remember now when you use that ability to sort of find a hole off this thing He he was a Roman s one. When this one times Oh, he is of us. Want to eat minus one? The saying goes, What is there? A point 34 for this? Horton is what 0.3 beautiful. And then the last one we have been asked to find a Scottish suspected Value's off. Excellent. So I will see that the expected very which can be written that this is the same as saying integrating the function over X And why? And then what? Spain? Because we are finding expedition off exit. There's no exit and find the eggs. Do what? So what? I am trying to say that this roofing can be retired. Sort of affinities are infinity's. The 0.1 e man is er five herbs is roping two way sort of eggs. The eggs. Now I'm just trying to interview this whole thing with respect to exit this world. Him respect it. But because you consider this can be integrated by using intervention by Pat. So we have anticipation. My parts here Integration. Bye. Cuts again? Yeah. Intubation buy parts to receive the Frenchness. What in the function is you the same Jules minus two bracket. Infinite to sign, Jim. Okay, so from this France and let our u equal to X So do you in the same sort of ex. Now let's our v equals what devi 0.1 e zero point five Axeman. Is there a point to? So we should integrate this one respect to really bad at the equal to 01 and 0.5 He man is there Open fires, X man It's What? So what do you do? The whole of this one is what we must find this function by. That's because we have it years What? 0.1 point five x My name is 05 x men is 0.2 way and then advances. What infinity to the minus the integration off this one. So you have affinity to settle And then we are zero wins one Is there a 10.5 e 05 accident Is that too then ex way now When you married this one, this hoping at infinity is equals. What? Zero And then Well, it's zero when excess infinity is the one This is zero is still zero. So we turned left with this one for us to integrate this woman with respect to X So it should be this um of infinity is zero. So we have a negative time already here because of this one. So 0.10 point five on an e. Zahra 50.5. Excellent. Zero point two way the ex So, yes, Trains who interviewed this one? Infinities are trying to integrate this respect. Ex. So this and this one must by this and this woman in any cheap. You're 0.0 opens fresh. So when you going to get this one? With respect to X haven't worked times one divided by money is 0.5. And if you have eaten your opponent's five excellent zero point wife or values infinity to zero. Do what now? When you certified the whole of this, you should be able to get you mean in signs There are shy to simplify. This opens one better weapons opened fire. The thing that says what? So what? We simplified the whole of this year. Negative zero points for right now one ness is infinity. This woods in your zero and one X when x this zero year this one he is a positive. And then we have your wife. Now when we try to integrate this one space in a way, we are also getting what my name is Two hostages, Manus. Zero points for and then we have one writer make it zero point two. Maybe he is that Wasim points Why? And then for violence ofthe way to eternity. Now you try to some fat cooler is we have zero for And you know that this got us one. So two and one, Why is infinity we have? And then when? Why is what? 01? Now, when you try to simplify the whole of this formula That nonsense word too soon This petition off eggs work too. Now we are also mean us to find the expectation off. Why? In tradition ofthe why So this petition off why us? The Seamus for Earth's is what infinity for all the sins ofthe way. And then nobody's off us if the function and then time sports. Why, Mary? And why the ex? So this world can be as infinity from zero and then infinity to zero 0.1. He man is 0.1. So's opened five extra minutes. 0.2. Why the s So we have our why already Days? Why can't come anywhere. Selim, Why here? All right soon. I don't want to integrate with respect to X face. When we had with X so This is a constant. This is a constant. So they can just come out ofthe bracket like that. Why? And then when you tried to interview this one with respect to X one on man is 0.5 e 0.5 X man. Is there a point to move away for values off as being infinity to zero And then we have Do you want now when you again shy, Just simplified the whole of this one. We have infinities. There are zero opens. One way divided by men is opens five and then when x is infinity. We have this on running to zero. And when excess zero we have, he gets zero point. And now again, this is also what intervention by path. So you can also see based on what we did That letter Are you equal to Why? So do you go to? Why not let our Davey you call to this hole function that is giving you see is what we are treating. This one is a constant. So I'm not going to bother with this ones who you have So point two why? And then the same assault. One brother, Burman, is open to. Is there a point? What? So I'm just bringing this one here, So the integration is going to be so 0.1, and then when is the 0.5 and then we have June times G, which is the same sort. Why? He's opening a way from the events ofthe xB, infinity to settle. And then we have infinitive too. So and then she was get affinity zero and one on menace to 0.2 way. Why? Yeah. So now when you try to simplify the whole destiny should be able to put a 20 something destined. You're this cool thing can also destroy. Provide us, man. Is there a point to? No. When s zero when Essence. What several. Here The answers were zero in Texas. Want someone s? Why is zero? The answer is one went. Why is infinity that asses sources there Also the whole of this moon go to survive and then we have a this then that. And then when you try to into get this one we're expecting Why again? You also get one by 0.2 time zero point to man is there or why infinity too, So yeah, So when you try to some fighting, cool off this and she will get 0.2 and then when is open, too zero for two times two? Yeah, And when Why? He sorts infinity about zero and went, Why is so we have one day. So when a certified hole of this and it should be equality zero point to develop by 0.2 which is also the same US two times, which is the same as what fight? Soon there has been a spectator off. Why is also the same issue?

To begin the problem. Let's write down the hypothesis. So we have the We have the density. If off X Come away. Is it going to for exploit when? Zero less than the contracts less than equal to one zero less cynical, wild, hysterical 10 Otherwise so. The first observation is, if off ex cla by unscripted, cynical zero, which is clear because extends widespread. And the Nicole do zero when X and Y lies in this range. And then we have to check the integration from they're 21 Consider 21 four ex white, Then do I. Followed by T X is equal to one. So let's evaluate this indiscretion and show that this is indeed a difficult one. So this becomes four 0 to 1 on the digression off wise y squared, divided by two extremes like squared divided by two 0 to 1, followed by DX. So four integration center to one x divided by two, followed by a DX. So this becomes too. At the integration off excess simplex squared divided by two 0 to 1 on this becomes one. So this very fight stuff second action for the density function. So if off X comma y this indeed on DDE. It should be e a density off ex sent away. So from this, let's find out the density off X. So the density off X It's going to be if capital X small legs set up too intricate over to other variable Sasebo to one for Excite dear I. So this becomes for X, followed by Y squared divided by 2 30 to 1. We just simply two weeks similarly to the density off. Why it's going to be two times five by the similar competition. Why we need this because they have to computer mean in part seat. So Fort Bard Be one we have to comfort. Probability exists Kate at the Nicole tohave if exoskeleton equal to have we have already computer density for X. So this is going to be half to one two weeks. T x so excess square because integration off Twix is excess square, have to one so one minus one over for just three over four. Next part p. Two. We have to compute probability off exists created the nickel to have and why's less cynical tohave. So this is going to be integration. Six escaped a technical toe have so have to one the why is less cynical to have. So they should be cedar to have four x y on d y t x. Okay, so then have to one on dhe this with the integration off. Why is simply white squared? Divided by two So x. Why squared? Divided by two. I have four and zero to half t x. I can cancel the two. So this becomes to one two weeks. If I put up our limit. This is simply one over four, right? So then followed by t x. I can cancel the two again. So this becomes too. So let's take the half outside. So this becomes half integration, their school back so half to one and then x t x. So no, this is excess squared, divided by two, half to one. So one over two. So one minus. If I take one over two square. So excuse me, that should be half because upper limit is half on the lower limit is going to be the two squared is four. So one of her eight, So this can be reaching us one over four one. Might us one over four. So one over four times three over four. It's according to three over 16 so that solves the part. B. Let's compute the mean so that me no fix. Let's call it X mean it's going to be the integration into Christian from 0 to 1 that density or fix. So let me write down the formula. Time. Six t x. So the integration for us from 0 to 1. The tense that Iwas Twix time. Six D x. Another integration off excess square is simply X cubed, divided by three clear to one. So this becomes to over three and similarly finally, Nick's. Excuse me. So why, I mean is equal to 0 to 1 so two times Why times, Why do I? It's the culture here at the same into crew as before, Only the X has been replaced by why so this because two or three as well on their salts. The probe

All right. So we've got this joint density function here and were we need to find the value sea. So we know that Ah, this entire density function over the region, eh? Of, ah f of x y z dee's equal to one Because it's a joint density function for end of variables. So we know that we know that one is equal Interval from zero to two to go on reserve two into going on a journey to of C x y z Deac steal I d z. All right, so that sea of the integral and ah, we can just do this as all these separate into girls because they're multiplying and one does not depend on the other. Okay, easy. Okay. And if we just look at one of these into girls X squared over to charity too, we see that that's able to to so that one is equal to C times two to the three. Two c is equal to one of the great. All right, so now we want to look at part be here, so part B, we're looking at everything less than or equal to one settle. We're into girl. Our shuttle into girls are all going to go from one toe zero one that way wth x y z d x y z And very similarly like our previous problem. For the previous part of this problem, at least we can waken you can do the same thing basically right One a into girls You're the one of X d. Jax into girls here to one of why do I do girl zero one of Z D And just looking at one separate one of these, the at one over to so that one over eight times one over two. Cute. It's one over twenty stuck two to the six she and over eight time whenever sixty four. Fantastic. Percy, that explains why Lizzie Lesson or equal to one. You won't find this probability. So we know Z is equal to one minus x minus y. And, uh, what else do we know here? Way. Know that Z is equal to zero. Then Y is equal to one minus X and ah, these are all bounded below zero. Next is around it or above, uh, extended by one. So we're doing this triple in general now, Okay? And one eighth x y Z First we do easy do I and DX eso This is a pretty standard straightforward into girl. So let's go ahead and do this out and ah x y times one minus x minus y squared. Do you know why Dia and ah, I'm actually in line to just leave this the rest of this to you? There's a lot of algebra here in integration, but it's pretty straightforward. It it's not that difficult. Okay, I will write a few of these. I guess you got zero one of one over twelve x to the fifth, minus one over three x to the fourth this one over to certain whenever two x to the third guy won over sixteen. Out here on there's one third X squared minus one twelve packs DX they And once you do this integration, you'LL get one over five seven six zero Ah, as are finally ends

Not as little question given about the joint BDF or is given to us in that question. So the joint bdf off x one extra and x three is f off x one x two x tree, which has given us. We ask e into x one x 21 minus x tree and this is for zero as an equal to X. Well, there's an equal one where again, extras als between the same values and same was with extremely such tight x one plus x two plus x tree are less than or equal proof one and zero. Otherwise so, uh, they've given us a value of key as 1 44 So the first part you it is too. Find out the joint video of X one and X trees. So the joint bdf off X one and extremely is someone finding this Aiken Day. Cool the tongue, which is given to us where function is only for X one and x three need extras constant here. So then the limits off integration will therefore change from, say, 0 to 1 minus x one because you can see that they have given extreme bliss extreme bliss experience less than equal to one. So many considered only extra next to me. So we can take the little dove, ext three as one minus x one. So the slimmer changes from 01 minus x one. And the function is skay into excellent into X two into one minus X tree. And we're integrating here. Attacks toe sex, too. They're running the joint. Pdf off x one and x three. So now, So the indignation is it takes two so that our terms are constant. So we keep Kay into X one into one minus x three as it is, and we indeed it extremes. We have extra square born to unlimited's from 0 to 1, minus x one. So if I put the limits than the function and getting here S k okay, since I'm done, the indication his limits will not come here. Yeah. So the final Valium getting here is gay ex one into one minus x three in one minus x. Wanda will square the whole thing divided by two. Next, just to find out the during video for for extra extra, I can just realized this final answer. Memphis, maybe See that therefore the joint. Pdf off X one and X tree is readiness if off x one x tree Which is equal. Do we have gay into excellent in the one minus x 31 minus x one Double square divided by do This is for the limits off X one between 0 to 1 and extremely also between the old one and zero. Otherwise, Rick, this is a party that have told us to find more over. The next part are we see is that we're supposed to find the probability even the rocks wolf types one and three together sneak up at most 50% off the sample. So when does the at most 50% it means that maximum is 50%. We can represent this in terms of variables as X one plus X tree will be max to max 50% which is 0.5 right. So when does the exam plus extras less unequal? Fight me. The limits are going from 0 to 0.5 for our X one and for ex leaders from 2.5 minus x worth. So then I can represent This hasn't tingled for X. You're taking 0 to 0.5 for extremely from 0 to 0.5, minus X one and the function that we have we've just found of all witches joined media for extra next three. So that's key X one in the one minus X tree, the one minus X one, the whole squared divided by two. And we're indicating forced with X three and then with X one. So to start with K upon to with the constant any so we can take that outside. So that's a ridiculous little golf 0.5. Then, uh, I can keep Kia born drew aside and we're integrating with extreme So x 11 minus x on the whole square will remain as it is in violation for one minus X three will be one minus. X tree worries to tow a phone to go official to fix three is minus funds. We divide here limit for this is from 0.5 minus x one with B X one now moving on to the next part. For that, we can write this as, uh, I'm getting in the constant outside So this changes toe minus one by four says from zero Do weren't five we have gay into excellent into one minus explained the whole square. And when I pulled the limits in the black and Tucker limiters one minus. This will change 2.5 plus x one. The whole square. Okay. And mine this one into the excellent as it is. So this is for simply five folder. We have minus one by four Indigo girls. You know. Boo 40.5 gained two x one. Now I'm going to expand this black it. You're so this genius toe one minus two x one less excellent square. And here, when minus 0.5, give me 0.5 classics from the whole square. So if I expand that black, it have alive 0.25 plus x one and a plus. We have explained the whole square minus one with the expert so further, I wouldn't want to play this excellent in this bracket here. So we have key as it does little dew 0.5. Then we have x one minus two. Excellent square messages X one Q into this back as it is. So going into five minus one is 10.75 negative points and five. So I'm writing this an excellent square no x one minus 10.75 The X one. Now we need to monitor play ball the brackets to perform the integration so there's a gate and giggle. Zero reporting five now expanding bore the brackets so I would get the terms as excellent. Square plus x one cube minus 10.75 x one minus two. Excellent. You minus two Ex memories before plus 1.5 excellent square. Plus we have X one race before less X one place for five and minus wind. 75 Excellent. Thank you in dilation is with the X one. So if I add in supplying though terms which have the same by Woz, we have minus key by four individual 0.5 expression. Finally, I get a sex families to five minus x one raise to four minus 1.75 x one cube plus 2.5 x one square Why in this 0.75 x one with B X one. Now if I integrate your the bones, I really Gators. There's my escape by Ford forming integration. We have expertise to six by six minus X memories to five by five minus 1.75 x Wendy's before by four plus we have 2.5 x one cube by three minus 0.75 x one square by two and the whole limiter from 0.5 Now, if you put the upper limit your mobile emitter zero enemy So we just need to put the excellent s mind, find the expression and get the answer. So if you value the whole part, then I'm going to get the answer. We're here by putting the value off. Key has won 44 because it's given in the question. So this is 1 44 by four in tow, the whole expression by putting the elements. So we have minus 0.5 days to six by six plus 60.5 days to fight by five. Then I have plus 1.75 point four into this is point. Fighting's to four minus 2.5 by three in 2.5 Q. Then I have less wind. 75 I do into we have wind file square right to the expanding Simplify the holding. The answer that we're getting here is 0.740632 So this is the answer for the party now moving on to the park. Si, uh, we were really good in the joint. Beautiful. Thanks for the next three earlier. So if we need to now find the marginal pdf off X one. So I would therefore say that the marginal pdf off x one is 1/4 ex one which is actually indignant off. Well, indignant. Now with experts, the limits of extreme will be from 0 to 1 minus x one. So the pdf Waske x one by do into one minus x 31 minus x one The whole square with the ex tree. So now we're going to integrate with ex trees. So the terms, Yeah. So I'm goingto keep gay. Expand upon you in one minus Excellent. The whole squad as it is. And intimidation from one minus x three will be one minus x three the whole square divide by coefficient minus one in tow and the liberties from 0 to 1 minus x one. So now if I pull the limits, I'm getting the Tom here as minus K by full X one into one minus x one the whole square. And if I put the limits. Here I get one minus one less x one, the whole square, minus one as this. So we can cancel one here and the final value by giving the minus outside Here in the bracket I can get the answer has kept on four x one into one minus excellent. The whole square into one minus x one squid. So I can Therefore I don't The marginal pdf s f f x one is equal to key by four x one one minus explained the whole square into one minus x one square. This is for the value of X one between zero and one and it zero otherwise.


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5 answers
[4 points] Determine the Lagrange form of the interpolating polynomial P(z) that interpolates function f (x) at x = 0, h; 2h, and 3h, where h > 0. (b) [6 points] Determine the quadrature formula for approximating I = Jc 3h f(c)dx that 3h results from approximating the integral by I ~ So P(x)dz.
[4 points] Determine the Lagrange form of the interpolating polynomial P(z) that interpolates function f (x) at x = 0, h; 2h, and 3h, where h > 0. (b) [6 points] Determine the quadrature formula for approximating I = Jc 3h f(c)dx that 3h results from approximating the integral by I ~ So P(x)dz....
5 answers
Question 3Which of the following Iines intersects the graph = of the function f(x) 4x+22y = 8x+2B,Y = -4x+2 c.Y = 4x - 2 0.5y = 2x -
Question 3 Which of the following Iines intersects the graph = of the function f(x) 4x+2 2y = 8x+2 B,Y = -4x+2 c.Y = 4x - 2 0.5y = 2x -...

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