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12. (6 points) The joint probability density function of X and Y is given by flx,y) = 0sxs1, 0<Y<2 Verify that this is indeed a joint density functionFind P(0...

Question

12. (6 points) The joint probability density function of X and Y is given by flx,y) = 0sxs1, 0<Y<2 Verify that this is indeed a joint density functionFind P(0.2 <X<1; [.8<Y <2)Bonus Problem (5 points)Let X and Y have joint uniform distribution on the region A. Let A be triangle with corners at (0,2), (2,0) and (0,0). Find El xlx=- 2

12. (6 points) The joint probability density function of X and Y is given by flx,y) = 0sxs1, 0<Y<2 Verify that this is indeed a joint density function Find P(0.2 <X<1; [.8<Y <2) Bonus Problem (5 points) Let X and Y have joint uniform distribution on the region A. Let A be triangle with corners at (0,2), (2,0) and (0,0). Find El xlx=- 2



Answers

Verify that $f$ gives a joint probability density function. Then find the expected values $\mu_{X}$ and $\mu_{Y}$ .
$$
f(x, y)=\left\{\begin{array}{ll}{6 x^{2} y,} & {\text { if } 0 \leq x \leq 1 \text { and } 0 \leq y \leq 1} \\ {0,} & {\text { otherwise. }}\end{array}\right.
$$

Well that's probably been given the following joint probability distribution. And I would like to first show that they are not independent now in order for them to be independent. So that's why must be equal to the product of the marginal distributions of the two. So in order to do this let's find the marginal distributions of the tune and from that are marginal distribution effects means we integrate out the why? So why does from 0 to 1 -X. So this is three X squared evaluated from zero. I'm sorry not three X squared be integrated with respect to why not with respect to X integrate with respect to why? This is to be six x. Y Evaluated from X0 To one -X. So this is six x Times one -X. This is our marginal distribution of X. Yeah. The marginal distribution of why we integrate out the X and two X goes from 01 Now we will integrate with respect to Act this is three X squared yeah Evaluated from 0 to 1. And so this will give us A value of three and so are marginal distribution for why It's just a value of three. Now clearly we can see. Mhm. That f of why why times F. Of X. XR two marginal distributions is not equal to our joint distribution. Thus that's why or not independent and that's what we need to show. Their army. Yeah on me We want to find the probability that that's better than a .3. What given that why is equal 0.5? In order to evaluate this this is equal to the probability That it's just larger than .5. And Why is 0.5 over The probability that why is 0.5 using a conditional probability laws? I'll talk there. That's just our joint distribution. And so our joint distribution is six X. That's where we get this on top and then on the bottom that's our marginal distribution are marginal distribution for why? Those are three. So this gives us this distribution here to be two X. Now we would plug in 0.5 for why? But there is no way to plug it into here. Mm We do want us to be bigger than .5. And so we integrate this From 05 to one because X has an upper limit one. So this is x squared evaluated from 0.5. To wonder That's why party on one With 41.5. And subtract This gives us .75.

Well that's probably been doing the following joint distribution. Now, the first thing we invite to find is the value of K and find the value of day. We know that the triple integral of K X y squared Z Must be equal to one. In order for us to a probability distribution. No extras from 0 to 1. Why goes from 0 to 1 and z It goes from 0 to 2. This means we have K times the integral from 0-2, brazil is easy. The integral from 0 to 1 of y squared. Dy the admiral From 0 to 1 of X dx Is equal to one. Now the integral From 0 to 2 of Z. Z is equal to The integral from 0 to 1 of why squared is one third And the integral from 0 to 1 Of that is 1/2. Yeah. Mhm. Mhm Two and one half canceled. So this tells us that K over three is one. Yeah, That's OK. Is equal to three. Yeah. Some things are joint distribution is really F of X. Y. Z. Mhm. This three X Y squared Z. With X&Y. Between zero and 1 Lindsey. Between zero and 2. That it helps us on B. Because one of the probability That acts as less than 1/4, Why is greater than 1/2 And Z is between one and 2. So finding this probability we want to integrate the triple integral of three at Weiss Birds. E Z goes from 1 to 2, Hind rose from 1/2 to 1 because it has an upper bound at one. Then Ash goes from 0 to 1/4 because it has a lower bound at zero. So this is equal to three times the integral From 0 to 1 4th versus the arts Times. The integral from 1/2 to 1 of Y squared dy Yeah, I'm gonna go from 1 to 2 of Z dizzy. So this is three times now. If we end great From 01 4th of X. Dx We get 1/32. If we integrate, why squared From 1/2 to 1, What do you 7/24? And then if we integrate Z DZ from one to We get three hands and then we multiply all these together. Mhm. This is 21/5, 12. Yeah.

In this question were given a joint density function And we want to verify that this is a first that this is a probability density function. Second, we want to calculate a few probabilities and third we want to find the expected values of both variables. We're assuming that both of the variables are discreet that they can take on any value. So how do we show that this is a valid pdf or a probability density function? We can do that by. We do that by making sure that the double integral over all of the reels. Yeah. Mhm. We need to make sure that this entire function Has an integral an area under this plane of one. So how do we do that with this question? Well that would mean that the that both double integral from negative infinity to infinity. I would have to be one. Fortunately We only have to go through one side because the function is only non-0 if x and y are both positive. So this means that what we actually have to check, we actually have to check the value of the integral from zero to infinity. Of the whole exponential function. Yeah. So you just have 0.1 times the exponential function of negative 0.5 X minus 0.2. Why? And thankfully in this case it doesn't matter the order in which you put your differentials. The integral can be split according to the according to the multiplication role of exponential. So have 0.1 times east, the -0.5 x D X plus the double integral Of 0.1 times e. to the -0.2. Why? Right? Yeah. And by this point you know how to do already. Mhm. We know how to do these integral is already and we actually only need one integral sign for each one. We know how to do improper integral. So evaluating each integral will give us You'll have 0.1 Over negative 0.5 times east, the -0.5 X. That goes from 0 to infinity. And the same thing. Yeah. Mhm. Mhm. The same thing there. Now the infinity terms when I plug infinity and will just vanish And this will just give us 0.1 times zero point time 0.5. Yes. Mhm. Plus 0.1 times one over. Don't forget these are actually being multiplied. I've made a mistake there. I made a mistake that there should be multiplied together because of the product rule for integral. So And because this is one we know that this is a valid probability density function. Yeah. I'll just write down the theorem that I used here. Okay, now the second part is we want to find two probabilities. You want to find the probability that why Our random variable y is greater than one. So that's the first step. Okay, how do we do this? Well all we need to do is we just need to change our bounds of integration a little bit. Yeah. So this will turn out to be the integral from zero to infinity of 0.1 times the X times E to the minus 0.5 X. Being multiplied by the integral from one to infinity of 0.1 times E. To the minus 0.2. Why it? And by similar logic We have that by similar logic to how it did the integral above. We have that this is going to be equal to approximately 0.8187. Yeah. Now the second part is we want that the probability mhm But our random variable X is less than two And at the same time I random variable why is less than four. That means that we're going to have our integral as going from 0-2 and from 0-4. And I can say that it's just going to set the beginning band will be zero because everywhere else the function is zero. So that will be the integral from 0 to 1. From 0 to 2 of 0.1 times each -0.5 x. Multiplied by the integral from 0 to 40.1 times E to the minus 0.2. Why? And doing these intervals in much the same way as we did before Will result in a probability of 0.3481. Yeah. Yeah. Mhm. Now the last part is we want to find the expected values of X and y. So we want mhm We want those values. So how do we find those? Well, the expected value of each integral. Yeah. Which will note down here, it's just the double integral over the real plain of why? Mhm. Yeah. And we'll have an X be multiplied by it. And the expected value for why similarly is going to be yeah the integral over the real plain of why times f of times the function mm So let's begin finding the expected value of x. 1st we know that this is going to be equal to X Will be the end score from 0 to Infinity and score from 0 to Infinity. And we'll have 0.1 Times X Times East zero just like that. Which we can again split into an integral from multiplication. Yeah. Okay. Yeah. In fact and the first integral we can do by parts and by similar logic. Yeah. Because of the first and then second integral. We can just do that on its own. Uh huh. The first one will require integration by parts but when we do so we're going to get that. This is gonna be too Okay. And similarly I'll just skip to the other to the part where we like to see it. The expected value for why It's going to be just 25 or that will be five Because of the same logic. What? And that's how you do this question

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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