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If X and Y are independent gamma random variables with parameters (a, and (B,A), respectively. Find the joint density of U = X +Y and V = XI(X + Y)....

Question

If X and Y are independent gamma random variables with parameters (a, and (B,A), respectively. Find the joint density of U = X +Y and V = XI(X + Y).

If X and Y are independent gamma random variables with parameters (a, and (B,A), respectively. Find the joint density of U = X +Y and V = XI(X + Y).



Answers

Suppose $ X $ and $ Y $ are random variables with joint density function

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





\begin{array}{ll}







0.1e^{-(0.5x + 0.2y)} & \mbox{if $ x \ge 0, y \ge 0 $}\\







0 & \mbox{otherwise}





\end{array} \right.$

(a) Verify that $ f $ is indeed a joint density function.
(b) Find the following probabilities.





(i) $ P (Y \ge 1) $










(ii) $ P (X \le 2, Y \le 4) $
(c) Find the expected values of $ X $ and $ Y $.

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

So for part A here one find a value of C, and this is called the joint density function. So we're going to integrate. So we know that if we integrate a new intensity function triple integral some joint density function, we're going to call this one F over the volume. It's going to be equal to one, because this is the probability. This is really what's going on here just to give you guys have been site. So we're interviewing from zero to infinity, zero to infinity here to infinity of C G to negative zero point five AKs was Europe went to Why was your appoint one z? You know d x d? Why Deasy? Okay, So to make this kind of formal, we're going to take as t goes to infinity. Okay, see to the negatives Europe, When's your five x he tonight? Observe Went zero to why? And e to the negative zero point one z d x d Y d z Okay, so you can pull out. This city has Teo comforting zero t Americans flipped this up according according each one doesn't depend on the other. Okay, so that this and now integrating this What do we end up with? Okay, so we're going to get, um e. And they use your one zero five x from zero to t her natives. You're up for it. Sighs great. And then e to the negative. Europe one zero two zero from Widens Your tea. Negative. Zero one zero two. So there were times in each one of these in the same thing over here. Me. So, um pretty easy, to be honest, to see that once we once we actually do all this multiplication now. Okay, so I see the men tears to infinity. No, for zero point zero five one over zero point zero side. He's going to be negative too. This guy will give us times Negative five in this girl. There was times native ten. No. And then what are we going to get here? We're going t night of zero point zero five t minus one times e to Naser of Missouri Fot won t minus one eatin a zero point one T minus one. So as long as t goes to infinity, each one of these is going to go to the girl. No. So we're going to get C times negative. Two times negative. Thought his negative one times negative. One times native one turns name and this is one hundred seed. We have to have this to equal to one, so C is equal to one over one hundred. That's the first. That's part eight now, Part B. We're integrating accident. Why? From zero to one. But Z from zero to infinity that are one over one hundred of E. Uh, the same thing here isn't sex. The t x t y easy. All right, so we know integrating with respects to see here. We know from the previous problem what we're going to get concerning one over one hundred lot. The constant zero, I guess we'LL do. This is a limit t goes to infinity Your tea eat in the reserve Want one's easy, easy And then we've got thes into girls here. Okay, so I'm not going to do this one out, But we can We can really just see that we're going to get ten. No. So that one will get ten and again. Not too. Not too difficult Teo, to see that. Yeah, And then for these guys in green well, these air pretty straightforward into girls Get times negative. E his negatives. Europe went five plus one over zero point five the negative e natives. You're want to plus one over zero point two. Okay, so, uh, doing all this out on you could use a calculator or answer. Um, it's going to be zero point zero seven one three two if you simplify your going again, end up with one linus heat in the desert winds or two. There was one money's eatin these aeroplanes, your five. And that's what their spirit. Yeah, that's, uh, Herbie and one more part part. See, while now we're just going to zero two one for each one. X, y and z one of the one hundred of our function. Remember, function is heat and NATO. Europe one zero five AKs for Europeans to why was your c dear x Do ID's. All right. So we got one over one hundred and we're going from zero to one. We need to make a jerk with five AKs. Detox zero one meeting a day because you're too high. Do I. Zero two, one. Eat unaided. Your one's easy. Easy. All right, so one over. One times one over zero point five negative. Zero point five. Doesn't matter you. Well, it's just easier. Point five and we're going native. You need to make a point. Five plus one one over. Sarah, point two eighty each. And negative. Two plus one, one over one. I'm getting eaten Later. Put one plus one. Okay. One thing that we see is that all these guys here were We're going to cancel out and our answers and be one minus D to the negative. Five one minus B to the native to one minus. Fees invaded. Kate. And this is about zero point zero zero six seven eight, okay?

To start the problem. We have touched your intensity. If off X comma y is it going to see X tents? One plus why zero less than equal to X, less cynical one and zero less cynical to wireless and equal to and zero other voice. The first part of the problem is asking to compute the value. Upsy because if it's a probability density function, so I know the integration from 0 to 1 that sex limit on the wild images from 0 to 2. That's C Times X times one plus Why so D Y d X is he going to one? And I can use this integration to compute the value for C salutes Evandro disintegration. So this in place see into questions there to one meta question. 0 to 2 on the interior. Integral. That exists constant. So this becomes X Times one plus. Why do you like D X is a good one. So this imply. See 0 to 1, because excess constant so ex. So this becomes Fight Plus y squared, divided by two. The LTD's from 0 to 2 DX. So let's evaluate this, but I do so this becomes too plus two squared, divided by two, which is four divided by two. So that becomes too. So this becomes four DX. So next I have four times. See integrations little to one. Let's go back. So I have X t X Is he called to one? So four time see this is going to fix the sky divided by 2021 is the one so four C times half physical to one. So this in place, see if he called 1/2. So that solved the first part of the problem. Let's move on. So next we want to compute the probability off excess list, and he called to one and why it's less than equal to one. So this is given by into question from 0 to 1, integration from 0 to 1. The 1st 1 is stakes. Live it on. The next one is the wind of it on dhe because we already know season going to have six times y plus one y plus one. Then do you like Deeks? Okay, let's take the half outside. 0 to 1 that sticks limit, and now this becomes X times y square divided by two plus, like on the limiters from 0 to 1 t x so half sitter to one. So this becomes X so half plus one t X. So this is just three over two and then multiplied by half. This becomes three or 44 one x t x. We just completed the decree, which is half. So Don Siri's three over four times half physical, too. Three over eight if you're going to three over it, so that stands there for part B so far apart. See, we're going to compute the probability off X plus y. It's less than equal to one Now. The thing Toe up service that zero list Any call to ex listen equal to one six can take a knee, fellas between 0 to 1. So if we fix X, then why can be off course? Why? Congrats. Minimum a zero but now exists Fix so I can get most one minus six. So that's going to be the wind of it. So if we sit set it up, it's going to be 0 to 1 that sticks limit and the wild images from 0 to 1 minor. Six. I want half, followed by extents. One plus. Why do you like DX Phyllis, evaluate this integral. So now this is going to be half integration 0 to 1 so x times y plus y squared to fight it by two. This is going to be 0 to 1 minor six dx so half liter to one so x times one minus six plus one minus six whole square divided by two t x hooking. So if we do the simplification, this becomes X minus X s Claire plus half one minus plastics a square minus two weeks DX and no half sear to one. So I have the constant Oh, I forgot to want to play this by six. So this followed by X So this becomes six minus X squared plus half or fix. Plus her for fix Cube minus. So to win this half cancels. So I have excess squared t X. If we simplify the expression, this becomes three or 42 X My head's to excess square plus half excuse DX Okay. Safely delivered the integration This becomes three or 42 exist squared, divided by two miners. Toe excuse. Divided by three plus half extremely for four, divided by four on from 0 to 1. So finally, three over. Four, minus two over three, plus one over it. On dhe. We forgot this half outside. We have to be figure, Lee. I want to play the entire expression by half. So this becomes faithful for 24 times. Half sickle to five, over four feet.

For F. Of X. To be a joint density function. Then we are saying that the W integral over out squared must be equal to one. So then we can write the double integral for the function to be equal to this. So then we have the W. C. Grassroots infinity. 02 Infinity. You have several points. Why? E experience my next 0.5 X. Blood. 7.2. Why have do I the X. So You can apply in this case here. So that this will be equal to 0.1 is a constance. So you have your W. C. Grant observed to infinity. There is an infinity. You have E experience, My name's 7.5 eggs Then E. Excellent -3.2. Y. You have the Y. The X. So this So we first integrates with respect to why? So you integrate with respect why and you have the cigarette. Zero point why we have this. Their point definately it's the power I guess user .5 eggs. The deaths game named Simba There to infinity Is the power negative 0.2 Y. The Y. So this would be equal to you have zero points one. This is negative 1.0.5. It's the power negative 7.5 X. The interval service to infinity. Then this are so also give us negative 1.02. E Experience -0.2 Y. The Interview. Uh 0 to Infinity. So what do we have? We have zero points one. This will give us 1.0.5. Then this also give us one Divided by zero points soon. Which is equal to one to hence verified is very frank. So we have been able to show that the double integral over the function must be equal to one since F of X. It's known zero. We kind for both X and Y. Which are no negative. We can find a probability Of why greater than or equal to one. This is equal to the double integral from negative infinity to infinity. Them from 1 to Infinity of F. Of X. Y. The Y. The X. So this is equal soon The insignia from 0 to Infinity. Then from 1 to Infinity of F. of X. Why the Y. D. Y. D. X. So this would be equal to the there it's infinity. You have 1 to infinity. You have zero points 1. It's the power -0.5 eggs, blacks, 0.2 Y. D. Y. The X. So this It's in the sense so we can apply the prophecy of in the 6th year so that we have zero points one is a constantly observe points one, the insignia there is infinity Of E. to the power negative 0.5 x. The X. Then the incident girl from once infinity. Yes. The poem zero. So why the way so this day will be equal to So you have this to be zero points one. This will give us -1 divided by 0.5. You have eaten to the power and it gets to 0.5 x. The inside of us are its infinity. Again we have negative one divide advisory point to is the poem negatives there? Point to why The limits that is from 1 to Infinity. So this we give us zero points one. Then this will give us one divided by 0.5. And this will be equal to one divided by 0.2. It's the power negative 0.2. And this is approximately 0.8187. It's an X seven. Then for I we find the operability so I they probably see that's our eggs Is less than or equal to two. Why less than or equal to four. So this would be because they see girl from negative infinity too soon. Then from negative infinity to fall of a function F. Of X. Y. Dy the X. So you have The insignia from 0 to 2 04. Oh zero points 1 is the power -0.5 x Plus 0.2. Why the Y. The it's so apply industries here and you have zero points 1 as a constance Have the senior from 0 to 2 of It's the power negative 0.5 x. The S. Then Then see aggressors four is the power negative zero point so why Dy so this will be He quotes to observe points one. You have this would be -1 divided by 0.5. It's the power he gets observed .5 x. The Interval 32. They negative one divided by zero. We have each of the power negative reserve point to why The Interval served 2 4. So this you simplify. If you simplify you have this is going to be equal to zero points one you have negative too. Each the pan negative one plus two. Then -5 ft. to the power negative 0.8. Last five so this will give me one. My next is to the 10.-1. They have one mine it's eat the pie negative zero point is and this is approximately zero points 3481. So this implies that's the probability of why these are that are equal to one. It's equal to eat the power negatives irv point to Which is approximately 0.81 87. And the probability that's our X. It's less than or equal to two. Why? As well as an equal to four as well it's equal to You have one minute. It's each to the -1. Then one minutes is to the zero point speeds. Which is also approximately 0.3481. For the parts be then the expected value for X and Y. So first for X. Which is X. Me would be equal to the doctor insignia. From negative infinity is infinity. You have X. The function F. Of X. Y. The Y. The eggs. The S. And this is equally observe. It's the infinity You have observed. The infinity of eggs F. Of X. Y. D. Y. The X. So this is going to be equal to is there is infinity, Deserved it to infinity. zero points 1. It's it's the power -0.5 x Plus 3.2 Y. The Y. The eggs. So we can apply indices here. The prophecy of indices So that we have zero points 1. As a constance. We have the c grand service to infinity of AIDS is the power -0.5 Kicks the S. Then then cigarettes always infinity Of its the power negative 0.2 y. The way so you realize that this parts for the first pass of our integration. This part is a product of two functions. So you apply integration by parts for this. So if you apply integration by parts This is what you have AIDS. You have zero points 1. This is going to give us negative one divided by 0.5 eggs. It's the power -0.5 x minus one divided by 0.5 You have eaten the power negative 0.5 x. So this is for this box they interviewed from service to infinity. Then if you integrate this part you get one divided by 0.2. So this day is going to be equal to zero points one. You have negative two X. It's the power -0.5 x. My next four. It's the power negative 0.5 eggs, 0.02 Infinity. This is us five. So you have zero points one oh and five which is equal to soon. So this is the expected value for X. Then we do it same for Y as well. Which is why I mean So I mean why I mean would be equal to the zebra observed the infinity. You have observed the infinity zero points 1. Why? It's the power gets in 0.5 it's Plus 0.2 Y. Dy the X. So let's simplify applying the purposes of indices. zero Points 1 is a constant out. We have DNC Grasser playing exerts infinity of it's the power Legacy Reserve .5 Eggs, the eggs. Then the in cigarette there is the infinity of why? It's the power negative 0.2 why? Dy So what happens? You realize that this part It's a product of two functions. To apply integration. Buy parts here. So this is going to be zero points when you have This interior will give us negative 1.0.5. It's the power gets 0.5. Yes. The interval service to infinity. Then this would give us negative 1.0.2. Why is the power ***? Is there a point to y plus one point. So this would be we have my necks 1.0.2 squared. They're going to square just like this box where we had 0.5 sq. So this part as well was just 0.5 Squared. So they have they're going to squared mm It's the power -0.2. Why? The intervals are too infinity. So this is going to be equal to preserve points one. It should be one divided by 0.5 in this backs is going to give us You get you five. Why? It's the power I guess he was there a point to why My next to 85 It's the power negatives are point to why the entire advisory to infinity. So we have 0.1. This will be soon. Then this would give us 25 which is equal to right? So then this implies that our ex meme that is X men is expected value of X which is soon then why? Why me expected value of why it's equal to five as a finer results


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