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We determined thatRy_' Yz) J 6(1 Y2), esewhere; <Yz < 1, valid joint probability density function. Find the margina density function for Y_fi(y1)J1where&...

Question

We determined thatRy_' Yz) J 6(1 Y2), esewhere; <Yz < 1, valid joint probability density function. Find the margina density function for Y_fi(y1)J1where<Y1Find the margina density function for Yz"Y2 J22VnereFindilv(Enter your probability as fraction:)Find the conditional density function ofgiven Yzfly 1lyz)whereFind the conditional density function ofgivenYi"f(yzlyi)where<Y1Find4|vi(Enter your probability as fraction )471 9Y2

We determined that Ry_' Yz) J 6(1 Y2), esewhere; <Yz < 1, valid joint probability density function. Find the margina density function for Y_ fi(y1) J1 where <Y1 Find the margina density function for Yz" Y2 J22 Vnere Find ilv (Enter your probability as fraction:) Find the conditional density function of given Yz fly 1lyz) where Find the conditional density function of given Yi" f(yzlyi) where <Y1 Find 4|vi (Enter your probability as fraction ) 471 9Y2



Answers

Let $X, Y,$ and $Z$ have the joint probability density function $f(x, y, z)=\left\{\begin{array}{ll}k x y^{2} z, & 0<x, y<1,0<z<2 \\ 0, & \text { elsewhere }\end{array}\right.$ (a) Find $k$. (b) Find $P\left(X<\frac{1}{4}, Y>\frac{1}{2}, 1<Z<2\right)$.

Here. We've been given that F of Y. One why two is equal 26? Yeah one minus Y two. As long as we're between zero and one. And we want to determine if they are independent or not. In order to do this, let's find both of our marginal distributions here. So if one of my one we integrate out the Y. Two. So why to goes from Y 1 to 1 six times one minus Y. Two. Yeah. And so this vehicle six times why two minus one half Why two squared? Mhm. Y two does from Y 1 to 1 just six times one minus one half minus search. Why one minus one half why one squared. And so this is equal to three minus six. Y. One plus three. Y one square. So this is our marginal distribution for F. Of Y. One. Let's find him for F. Of Y two. Mhm. F one F two of Y two is equal to the integral of six. There's one minus Y two. Do you want to? And I look at what Y two goes from. I'm sorry look at why why one goes from it should be why one here we're integrating out to Y. One why one went from zero? So why to? Yes. And so now we integrate and this is six times one minus Y. Two as times why one? That's why one goes from zero to Y two. And so this is set times one minus Y two times Y to this is our marginal distribution for F two of Y two. Now F one of Y one times F two of why two? We said F one was three minus six, Y plus three Y square thanks. We said F two is six times one minus Y. Two times. Why too? Well clearly when we multiply these, this will not give us our original six times one minus Y. Two. It was F of Y. One why two. And so since F one of Y one times have two of Y two does not equal F. Of one why two? This tells us they are not independent, not individual.

Yeah that's probably been given a falling joint distribution function. I would like to begin by finding the marginal distributions triple X and Y. And the marginal effects just means to integrate out the why? As we integrate from 0-1, three X -Y. over 11. The why? And so this becomes 1/11 times three X y -1/2 White Square. His wife goes from 0 to 1. We put in one for why this is 1/11 Times three X -1/2. So this is our marginal distribution for X. And then for why are marginal for why? This means integrate out the Y about the ex. And so extras from 1-3. So we integrate from 1-3. The three X minus Y over 11. The yes. So this is 1/11 times 3/2 x. where minus Y X. Evaluated from X. is 123. So this is 1/11 times 3/2 times three square minus three Y -1/11 Times three. Have sometimes once where -Y. Times one. And when we combine like terms and simplify this tells us that are marginal distribution of why there's no you have to Y plus 12 all over 11. So that's our marginal for a while now would be we would like to know if X and y are independent and B is a simple no. And that's because we can clearly see that fxfx doesn't I'm sorry, that expects mhm times F. Y of Y does not equal our joint F of X. Y. So for that reason we know they're not independent on C. We want to find the probability that X is greater than two. Now here, all we need to focus on is the marginal for acts And so it's 1 11th Times three X -1/2. And so we just need to integrate this From 2 to 3. Remember X has an upper limit on 3? We don't need to look at the joint, we just need to look at the X. And so we integrated here. He was this 1 11th Times 3/2 expired minus one half X evaluated for Mexico's too just three. So now we're just going to plug in three Party into and then subtract. I don't know if we do. This gives us a value of seven over 11.

Yeah. It's probably want to find the marginal distributions given the joint probability distribution here. Now in order to find a marginal distributions, we need to integrate out the other herb. So F X with X. It means we integrate out the Y. That's we integrate from 0 to 1 three x minus Y over 11. Do you want? And so this gives us 1/11 times three X y minus one half. Y squared evaluated from y is zero to one point in 1.10 for Y. And then some track this gives us 1/11 times three x minus one half is our alphabet. Okay. Now, similarly for F some Y of Y we integrate out correct. And so this is 1/11 times three halves that squared minus X. Y. Evaluated from articles 123 Yeah. And so now for actually going to plug in one, we're going to plug in three and then we're going to subtract them. And whenever we do this gives us 1/11 times 12 minus two. Y besides our marginal distribution. For why? Mhm. No. And be we want to discuss that these are independent but clearly we can see that our joint distribution is not equal to the product of our marginal distributions. In order for them to be independent, the joint distribution must be equal to the product of the marginal distribution. And so since this is not true, they are not independent. No. And see we want to find the probability that X is greater than two. And so this is just the integral from 2 to 3 of just our X distribution. We do not need the Y. We just need the X distribution. And so we'll integrate from 2 to 3. This marginal that we just found, which is 1/11 times three X minus a half. So our anti derivative here, it will be 1/11 times three has X squared minus one half X evaluated from access to 23 And so then now we're going to plug into 40 in three and then some traveling. And or if we evaluate this sonia was seven overall up. Yeah.

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


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