5

8) The joint and marginal pdf' s of X= given by amount ofalmonds and Y = amount of cashews were flx,y) {24y 0sxs1,osysi,x+ys1 otherwise fxG) = {12x(1-x2) 0 s*5...

Question

8) The joint and marginal pdf' s of X= given by amount ofalmonds and Y = amount of cashews were flx,y) {24y 0sxs1,osysi,x+ys1 otherwise fxG) = {12x(1-x2) 0 s*51 and otherwisefrly) (12y(1 - y2) 0 <ys1 and otherwiseFind Cov (X,Y)= A)-2175B) 2/15C) 4/25D) 2/5

8) The joint and marginal pdf' s of X= given by amount ofalmonds and Y = amount of cashews were flx,y) {24y 0sxs1,osysi,x+ys1 otherwise fxG) = {12x(1-x2) 0 s*51 and otherwise frly) (12y(1 - y2) 0 <ys1 and otherwise Find Cov (X,Y)= A)-2175 B) 2/15 C) 4/25 D) 2/5



Answers

The joint probability density function of the random variables $X, Y,$ and $Z$ is $$ f(x, y, z)=\left\{\begin{array}{ll} \frac{4}{9} x y z^{2}, & 0<x, y<1,0<z<3, \\ 0, & \text { elsewhere. } \end{array}\right. $$ Find (a) the joint marginal density function of $X$ and $Y$; (b) the marginal density of $Z$; (c) $P\left(\frac{1}{4}<X<\frac{1}{2}, Y>\frac{1}{3}, 2<Z<3\right)$ (d) $P\left(0<Z<2 \mid X=\frac{1}{2}, Y=\frac{1}{2}\right)$.

Yeah, that's probably been given the table and we would like to use this table in order to find the marginal distributions for both acts and why? We'll just begin by finding the marginal distribution franks Notice that actually takes on two different values. X takes on two and 4. So this means the death perfect is going to be a piecewise function With two different values Only taking on the values of two and 4. In order to find the probabilities, add up everything in the to column for eps. So add up all those different probabilities. I appoint one Plus .2 plus .1 0.1 plus 0.20 point one. That was everything in the X equals two columns. And adding those all together gives us 0.4. So the probability access to 0.4 now we do a similar thing for four. We just add up everything in the four column and when we add those all together It gives us 0.6. And so the probability that X is four 0.6. And so this is our marginal distribution tracks now for why we do the exact same thing except now we'll be adding up columns. Mr I'm sorry. Now we'll be adding up rows rather than columns. Yeah. So why it takes on the values of 1, 3 and 5? Mhm. Okay. And so let's add up everything in the one row for why To everything in the one rule for 3.1 plus .15 .25. So the probability wise one is 0.25, Okay .2 plus .30 means that the probability of Y is three is 0.50. And then lastly adding everything up in the five column for why Does this .25? And so here's our marginal distribution for what?

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.

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?

So this question we have been given that this Annie's arrival Damn analogies that every time some digging access, uh, Chinese. But every time I'm getting why, as, uh, Ivy's But I have a dime. Okay? And if the older that accent by so uniformly distributed on the double five coma six. Okay, so to start with the first part, what they've lost, we've been given the joint pdf off x and y. And based on that, we're supposed to write down the, uh, the marginal pdf sand the joint BDs. So here I can say that of the joint BDF pdf can be off laying eggs and why you can run down this as productive off marginal. Pdf ce off Ex sent by And this were saying because both x and y are independent evened. So since they're independent, I'm going to use this formula and forced right down the marginal PDS. So first I'm writing for next. So we safe effects is since the uniform distribution does one and zero. So just one between the double 5 to 6 as forgiven question and it is zero and sweat, and similarly, for my also this will be one and zeal for five between, but I also between five and six, and it is zero otherwise so, combining them, we can ride out the joint pdf that therefore, the joint pdf is if off x comma y can be denies. This is one for the conditions off excess exit between five and six and, uh, similarly, why is also between five and six and Tzeitel answer So this week Britain is the joint belief of X and bite. Next part is too cancel. It would be when we say the probability the board arrived between 5 15 and 5 45 So to find this probability, we got to write this in terms of X and buying doubles. So which represents as excess between 5 15 on 5 45 Similarly, they have I also between fine 15 and 5 45 Now, since the diamond over of you converting this into a number so we can convert Ah 5 15 to 1 number by, uh, taking this as 5 15 minutes, five hours so into edo, 60 minutes to convert it into minutes. And plus this is 15 minutes, right, and we can divide descent or two by 16 minutes to again converted into a number. So which actually gives me 5.25 And in a similar way, 5.4 fire also lettuce converts. This is five hours into 60 minutes. Plus we have another 45 minutes, and the whole thing is a 1,000,000 divided by 60 minutes. So in terms of numbers, this will now change to five point 75 Which means that indirectly, now this probability, which I'm trying to find I can write this as remember Eddie or for using the same product or swooned over the multiplication of individual except by values. But the values are changing from 5.255 point 75 So we take. This is 5.25 less and equal to X. There's an equal to 5.75 and in tow. Probability off 5.25 Lesson equal. Vilas unequalled If I 0.75 So if I multiply these two values, then huh I asked for the definition. The integral for X and by will be given by India. Elation off one because the function is one but the limiters from five point through fire to 5.75 and into the next part is integration off one device with the same limbs? Yeah. So if I indicate, then we're getting the value here as X from 5.25 to 5.75 and in tow. The next is by 5.255 point 75 So we get this value as 0.5 into a 0.5 and so you're 0.25 So this is the heartbeat, and we've just completed now to draw the part. See, I'm Wolf was considered or diagram where we can find out the other details from the diagram. So let me just say, like the dying, I'm forced. And it is just be sciences. Diarrhea? Yeah, So you can see. Then we've drawn two lines which represents the line. First is why minus X is equal one by 60 others X minus y is equal one basics along with the coordinate points that we have brought here and we have shaded the triangle regions also based on the values that are given to us. So now from this diagram, you can see that the points are taken in minutes and writing here that the points, huh? Taken in minutes originally. Okay, but in equation, when we're solving it has to be an hour. So I'm saying, but an equation, it is in powers. So because of this now I'm going to find out the required probability. So we say therefore, the require probably can we agree. Written as this notation we have given, the question is more off. X minus way is less than equal one by six you saw in the graph of plotting these equations and now you can see that this can be given by one minus probability off would the triangle So which is Triangle ABC And the other is ah, strangled the f So this will be one minus provocative. ABC will be the values that we have that it's half in tow. The other value is fine. By six into five by six. This actually represents the idiot who would happen to base into height minus here. It'll to be 1/2 and do five by six into five by six Simplify! I'm getting this answer as you 0.30 Firefight


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