5

Let Ho(y, 2) be the following bivariate distribution 1 - e-y _ e-z 5 e-(y+z+Oyz) , y 2 0, z > 0 Hol(y, 2) = 0_ otherwise, where 0 € [0, 1] is a parameter W...

Question

Let Ho(y, 2) be the following bivariate distribution 1 - e-y _ e-z 5 e-(y+z+Oyz) , y 2 0, z > 0 Hol(y, 2) = 0_ otherwise, where 0 € [0, 1] is a parameter What are the marginal distributions of Y and Z?

Let Ho(y, 2) be the following bivariate distribution 1 - e-y _ e-z 5 e-(y+z+Oyz) , y 2 0, z > 0 Hol(y, 2) = 0_ otherwise, where 0 € [0, 1] is a parameter What are the marginal distributions of Y and Z?



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 $P\left(X<\frac{1}{4}, Y>\frac{1}{2}, 1<Z<2\right)$.

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.

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?

This problem. We've been given the following joint distribution for the variables X, Y and Z. Our first task here is to find the marginal distribution of X and Y. Okay, now we're going to use the fact that these can be shown to be independent in order to break this up and so notice that we can write this as four X. Y times 19 Z swing. Yeah. Now there's a reason I broke the four nights up in that manner. And so that's so if we integrated Z from 0 to 3, that would give us a value of one. And similarly, if we integrate four X. Y over this entire square where both X and y are between zero and one, then a role for X. Y would also give us one. I have been broken apart like this. This tells us that are marginal distribution of action. Y. Mhm. Is able to four X. Y. We're both X and Y. Or between zero and what Now someone really on me where we want the marginal distribution of Z. That's just this other piece out here. This is 1/9 Z squared. Or Z is between zero and three. This is our marginal distribution breezy now and see we want to find the probability of X being between 1/4 and one half. Yeah. Yeah. Uh huh. Okay. Yeah. Of why being greater than one third problem and Z being between two and through. And so we need a triple integral here with extra point from 1/4 to 1 half. Well I going from one third to one because it has an upper bound at one and Z going from 2 to 3. Mhm. Mhm. I'm going to do this is three different intervals because we can break this apart. We can pull the four nines out front and then have the integral from 1/4 to 1 half of X. Dx. Uh huh. We have the integral of one third to one of why do why in the interval from 2 to 3 of Z squared dizzy again just breaking this entire integral part as we have the four nights out front and then we're gonna integrate X from 1/4 to 1 half. And then I will give us a value of 3/32. We're going to integrate why? From one third to one giving us a value of four nights. And then we integrate Z squared from 2 to 3 which is the value of 19/3. And so then we multiply all these together and we do this gives us the value of 19 over 162. Okay. And lastly on C we want a condition R. And D. We want a conditional probability what is the probability the Z. is between zero and two given that both X and Y. Mhm. Are equal to one half. Mhm. Now I'm finding this probability this will be the joint distribution at one half. One half. And then Z going from 0 to 2. All over the marginal of X. Y evaluated at one half. Yeah. Now we get in with the top there, the joint distribution of F. Of one half, one half and Z going from 0 to 2 it means plug in one half for both X and Y. I'm leaving us with 19 Z square and then we'll integrate this from 0 to 2 because we want Z to be between zero and two evaluating the central. It was 8/27. Okay. For the bottom we need F of X Y evaluated at one half one half. So it's a joint distribution that we found right here. We take four times one half times one half, diminish the value of one, so that's one. And so this means that we needed mhm. 8/27. All over one. Which is it paid over 27?

Yeah. This problem will be given to probability densities. The first is G. Of X. It's 24 over X. To the fourth or X is greater than two. Yeah. And the second is H. Of Y. For each of wise to why with the Y being between zero and one now we are told that these are independent. And so this means that our joint distribution F of X. Y. Is the product of these two, says it's 24 over at the fourth times two. Why? Which is 48 X to the negative fourth. Why that is our joint distribution. And notice that acts is greater than two. And why is between zero and one. Know what we would like to find is the expected value of Z. Which is the expected value of X. Y. Mhm. No. This is going to be the double integral. That's why times or probability distribution sometimes 48. Excellent and so forth. Why? Yeah. No extras from two to infinity. And why it goes from 0 to 1. Yeah. And so this is april 2. 48 times the integral from 0 to 1 of why times Y. Is y squared do Y? And then times they entered all from two to infinity of X times X to the fourth which is excellent, negative third D. X. And so we just broke that integral into those two different parts of. Once we have this this just turns into evaluating each of these different intervals. So the first thing we have is that 48 out front. And so the 48 will just say his name. And then we have one third y cubed evaluated from y zero to one. Then we have negative one half X to the negative second evaluated from access to infinity. So this is 48 times for one third white cube evaluated from 0 to 1. We just plug in one, plug in zero and then subtract, giving us one third negative one half X the negative second evaluated from two to infinity. We're going to take the limit as we go to infinity and then plug into and subtract those two values. This is gonna give us 18 So now we have 48 times, one third times and eighth, which is to.


Similar Solved Questions

5 answers
Assume that scores on the bone mineral density test are normally distributed with mean of 0 and standard deviation of Then For & randomly selected subject find the probability of a score between 1.37 and 2.42 Find the score separating the lowest 75% of scores from the highest 25%.
Assume that scores on the bone mineral density test are normally distributed with mean of 0 and standard deviation of Then For & randomly selected subject find the probability of a score between 1.37 and 2.42 Find the score separating the lowest 75% of scores from the highest 25%....
5 answers
Polymerization what he got is the ) . If someone conducted an experiment on step erowth sealed container that wa; under & concentration of the monomer at differemt time in whether stong constant temperature and pressure. Please discuss how You will dctrerpiro for the process! Mf yes; ho aS the catalyst Can you determine the value "k' acid was used will you determine the value ofk?
polymerization what he got is the ) . If someone conducted an experiment on step erowth sealed container that wa; under & concentration of the monomer at differemt time in whether stong constant temperature and pressure. Please discuss how You will dctrerpiro for the process! Mf yes; ho aS the c...
5 answers
0 211 0 1 1 IU ==Toniting81FK Queston [ 8
0 2 1 1 0 1 1 IU == Toniting 8 1 FK Queston [ 8...
5 answers
How is hydrogen sulfide generated in the laboratory?
How is hydrogen sulfide generated in the laboratory?...
1 answers
Find the exact value of each expression. Do not use a calculator. $$ \tan \frac{\pi}{4}+\cot \frac{\pi}{4} $$
Find the exact value of each expression. Do not use a calculator. $$ \tan \frac{\pi}{4}+\cot \frac{\pi}{4} $$...
5 answers
Many multidomain proteins apparently do not require chaperones to attain the fully folded conformations. Suggest a rational scenario for chaperone-independent folding of such proteins.
Many multidomain proteins apparently do not require chaperones to attain the fully folded conformations. Suggest a rational scenario for chaperone-independent folding of such proteins....
5 answers
Cobalt(Il) ion concentralionWnal- solulion prepared b} mixing 415 mL o/ 0.445 M coball(II) nitrale with 425 ml . OF 0.244 M sodium hydroxide? The Ksp Ol coball(II) hydroxide[col+|
cobalt(Il) ion concentralion Wnal- solulion prepared b} mixing 415 mL o/ 0.445 M coball(II) nitrale with 425 ml . OF 0.244 M sodium hydroxide? The Ksp Ol coball(II) hydroxide [col+|...
5 answers
Determine by direct integration the moment of inertia of the shaded area with respect to the $x$ axis.
Determine by direct integration the moment of inertia of the shaded area with respect to the $x$ axis....
5 answers
NoTEshOlenanngnnquatiorJmhinman conuilon9}
NoTEs hOlenanngn nquatior Jmhinman conuilon 9}...
1 answers
In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of $g(x)=\log _{2} x,$ and (b) graph the function. $$f(x)=-\log _{2}(-x)$$
In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of $g(x)=\log _{2} x,$ and (b) graph the function. $$f(x)=-\log _{2}(-x)$$...
5 answers
Bird wings and bat wings are considered to be ___.a. analogous and homologousb. analogous and homoplasticc. analogous, homologous, and homoplasticd. homologous and homoplastice. homoplastic
Bird wings and bat wings are considered to be ___. a. analogous and homologous b. analogous and homoplastic c. analogous, homologous, and homoplastic d. homologous and homoplastic e. homoplastic...
1 answers
A $0.025-$ g sample of a compound composed of boron and hydrogen, with a molecular mass of $\sim 28$ amu, burn spontaneously when exposed to air, producing 0.063 $\mathrm{g}$ of $\mathrm{B}_{2} \mathrm{O}_{3}$ . What are the empirical and molecular formulas of the compound?
A $0.025-$ g sample of a compound composed of boron and hydrogen, with a molecular mass of $\sim 28$ amu, burn spontaneously when exposed to air, producing 0.063 $\mathrm{g}$ of $\mathrm{B}_{2} \mathrm{O}_{3}$ . What are the empirical and molecular formulas of the compound?...
5 answers
(15) Sketch the curve using the methods of section 4.5. Y=x+2xls domain? Asymptotes? intercepts? symmetry? intervals increase/decrease? local max/min? concavity, inflection points?
(15) Sketch the curve using the methods of section 4.5. Y=x+2xls domain? Asymptotes? intercepts? symmetry? intervals increase/decrease? local max/min? concavity, inflection points?...
5 answers
Tdne HuThe region D shown belowcan be written in polar coordinates as D = {6.011<r<2 {<0< {1Select one: True False
Tdne Hu The region D shown below can be written in polar coordinates as D = {6.011<r<2 {<0< {1 Select one: True False...
4 answers
2. Find the determinants of the following triangular matrices_ [ -[ 0 3 10 06)5 2 1 7 8 8 3 5 2
2. Find the determinants of the following triangular matrices_ [ -[ 0 3 10 0 6) 5 2 1 7 8 8 3 5 2...
5 answers
Pea = Youinrott Kionnispalltmm bndca tulhIniual bpuedalan aralouerius ubOruhontonlalJouthan Wrotbasoba | Irom tnal briage walh Ineaal eonado13m473 9 0/ 30 dgoharconll HcotinartalancaTra lemughallnamantWoernlnntthtlnmeKeennllnaMc I urluec Mlertlantaebaanall rench tna sme Tuimer nuiahiThutunru balandino bnttlEnm EmecIhntt Fechant conts
Pea = Youinrott Kionnispalltmm bndca tulh Iniual bpued alan aralo uerius ubOru hontonlal Jouthan Wrot basoba | Irom tnal briage walh Ineaal eonado13m 473 9 0/ 30 dgo harconll Hcotina rtalanca Tra lemughall namant Woernlnntt htlnme Keennllna Mc I urluec Mlert lantae baanall rench tna sme Tuimer nuiah...
5 answers
Question 3 Not yet answeredGiven 2 +IncX) =e then dy dxMarked out of 2.00Flag question2y x(y+2)X(2y+1)V2y+1) 2y xly+1)
Question 3 Not yet answered Given 2 +IncX) =e then dy dx Marked out of 2.00 Flag question 2y x(y+2) X(2y+1) V2y+1) 2y xly+1)...
5 answers
1 Il-nelther Determine [PARAlELror 1 perpeedicular 32 + 1 aibue 431 the PERPENDICULAR, etiteen planes 1 are parallc l Jnoa 1 Jjmsur 1 one declmal place, 7 plane8perpendlcular;
1 Il-nelther Determine [PARAlELror 1 perpeedicular 32 + 1 aibue 431 the PERPENDICULAR, etiteen planes 1 are parallc l Jnoa 1 Jjmsur 1 one declmal place, 7 plane 8 perpendlcular;...

-- 0.021013--