Let V=XYFind the joint probability density function of (X V) fxv(xv) Sketch the support of (X, V)ORFind the joint probability density function of (Y, V) fyvlyv) Ske...

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

So now we have the density function ABBA backs y equals 24 x y for X is 0 to 1. And why in 0 to 1 zero are the words the expectation with respect to X and with respect, Why is defined as the in general are the density times adds the eggs and did this city. That's why do you Why? Supporting over data we're gonna had for this one for excessively four white downstairs who went X squared P s which use or three. And for diversity with respect. Why we have four x reserved one West square D y and that's gonna be four x over three.

Okay, so now we have the joint density function. After That's like was 23 halves times x squared plus y squared for X. Is your one included on Why same and zero. Otherwise, we're gonna find they expected value with respect to X and with respect. Why the density function this symmetrical? So, um, the numerical value for you, axe and new I will be the same. We don't need to take it with one of them and the other one we just, uh we get a result from the symmetry. So formula vats we have 010 to 1, which is the boundary of the axe And why we have ex constant density three halves times X square plus voice where in the XY y so were first developed it with respect That's we're gonna have three over to execute. Who, us? Let's move the constant over here X y squared the x d y. So this equals 2 3/2 0 to 1 extra before before plus two, um, X squared over two times y squared from 0 to 1. Do I? So this is three over 20 to 1. We have 1/4 who us half was squared. Do you? Why? So the answer is the answer for that is why were four plus why Cuba Over six from 0 to 1. And the answer for this is through or two times file or itself so is eight over five, and by sentry, we have mere for in close to five or eight.

You know there's probably been given the following probability distribution and I would like to find the marginal distributions not to find the marginal distribution for X. We need to integrate out the winds were entering out from 0 to 1. X plus Y. The wife. And so this gives us X. Y plus one half. Why squared evaluated from Y is 0- one. Mhm. And this is X plus one half. That's our marginal distribution threat. This acts plus 1/2 What? That's going from zero 20. Now for marginal distribution for what I notice that this is just going to be the exact same thing because you do the exact same mineral. So similarly we have F. Of why is why what? A half L&B. We want to find the probability that X is bigger than .25 and yeah why is greater than a half? Mhm. So here we need to integrate X. goes from 0 to 0.25. And why will then go from I'm sorry, extras from 0.25 to one. We have an upper amount of one. So it's 0.25-1. And why does from .5 to 1 about plus? Why the why? I'm sorry. Dx and Dy so integrating with respect to act first This is the integral from 0.5 to one of one half. Have squared plus. That's why evaluated from x 0.25- one. Do you want? Yeah. Uh huh. Yeah. So here's what we're going to do is plug in One party in 0.25 frets. That's attractive. And so this will have the interval from 0.5 to one Of .75. Y Plus .46875. Do you want? Now we integrate this and so this gives us, yeah, whenever we integrate .75 y plus .46875, we end up with 0.375 Y squared Plus .46875 Y, Evaluated from Y is 0.5- one. So then we plug in .5, we plug in one voice attractive. This is .5156- five.

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