5

Queation 02: harmony for the success of the ofa missile system" work electrontes components components (X: for first Tiko denote the life In years ofthe two_ t...

Question

Queation 02: harmony for the success of the ofa missile system" work electrontes components components (X: for first Tiko denote the life In years ofthe two_ totalsetert LXund Y t joint probability of X and Y is component Yfor the strond one) The 0 <* $2 0 s" $4 "" f) = ~lsewheremarginal - density functions for both random variables Give the the life of the first component exceed year and the life of b) What is the probability that the second one is between and 3 years? De

Queation 02: harmony for the success of the ofa missile system" work electrontes components components (X: for first Tiko denote the life In years ofthe two_ totalsetert LXund Y t joint probability of X and Y is component Yfor the strond one) The 0 <* $2 0 s" $4 "" f) = ~lsewhere marginal - density functions for both random variables Give the the life of the first component exceed year and the life of b) What is the probability that the second one is between and 3 years? Determine whether the two random variables are dependent or independent?



Answers

Two electronic components of a missile system work in harmony for the success of the total system. Let $X$ and $Y$ denote the life in hours of the two components. The joint density of $X$ and $Y$ is $$ f(x, y)=\left\{\begin{array}{ll} y e^{-y(1+x)}, & x, y \geq 0 \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Give the marginal density functions for both random variables. (b) What is the probability that the lives of both components will exceed 2 hours?

Hello. Even I just even question the lifetime off. Two components in a certain device, our random variable sex and weigh on. We have the probability distribution function. For this we have to find out the real lady that bought the components function for at least 12 months without failing. Thanks. It is given that components should functional for at least 12 months so we can write X Plus y is greater than a request to 12. And we should also and we should also have works plus way less than I go. So 48 right, so we can find the shared edition. The shared edition will be similar to this thesis. The 48 and 24. Right. The line is X plus wise rather than request to 12. So and both should also request to 12. Both components should work without felling. Right. So this is the question Why is the cost of well and access it cost to 12. So this is the region that we want, right? The shared region is between X rated and it goes to 12. Why get? And then it goes to dwell and do X plus y less than I got to 48. Yes. This is the definition for the shared A vision, right? So we have to find B. Actually, it another cost it well and like that. An inquest heard. Well, it is equal to integration from 12 to waiting and 12 to 48 minus x one by 92 one by 9 to 16 48 minus two x minus y dx dy y like we have to simplify this integration. So when you simplify this, we get the last result. This 1800 x minus 48 X squared, minus started. Six Axis Square plus four x Q by three plus one by 12 48 minus two X cube. So 12 to 18 like so When we simplify this, we get it has won 44 by 9 to 16 which comes out to be won by 64. Right, So this is the probability

We're told that a system has two components. Joins pdf of the lifetimes of the two components and system is given by f of x y equals C times 10 minus X plus y for X greater than zero. Why greater than zero and X plus y less than 10 Rexene wire in months in part? A. We're told that the first component functions for three months and were asked to find the probability that he's second component functions for more than two months. Well, in this case, you want to find probability at the second component, which is why the length of that is going to be greater than two. Given that first component lives a lifetime X of length three months, X equals three probably. Why greater than two? Given the X equals three. To do this, we want to first find the marginal probability density function of X and then form the conditional probability density function. So we have the marginal probability density function of X is the integral over the possible values of X. So from zero or any growing, the possible values of why so from zero up to 10 minus x. This is because experts. Why has to be less than 10 um f x y, which is C times 10 minus X plus Why de y which simply going to be well, we can factor out the sea and then integrating. We had 10 times 10 minus X. I guess you could say 10 minus X times 10 minus x So 10 minus X squared and then we have in to grow. Why this is just going to be native. Why squared over two? So we have yes minus 10. See Time's 10 minus X squared over two. She was at the BC Times 10 minus x weird over to and therefore it follows that the conditional probability density function. F y given x of why given x. This is equal to f of x y over FX FX, and this is see times 10 minus x plus. Why over see Time's 10 minus X squared over to which can be written as two times 10 minus x plus y over 10 minus X squared for X greater than zero. Why greater than zero? And in fact, why has to be greater than zero in less than 10 minus x? Finally, we have that if the random variable X equals three, then it follows that there in invariable. Why? Well, this has to be less than 10 minus three, which is seven. So it follows that the probability that why is greater than two, given the X equals three, this is mhm, integral from y equals 2 to 7 of If. Why, given three de y and substituting we have. This is the end to grow from 2 to 7 of two times 10 minus 10 minus three, which is seven minus. Why, over 10 minus three is seven squared. So over 49 do I. This comes out to be 25/49 which is approximately a decimal 0.512 just over 0.5. Then in Part B, we're told that the system continues to work on Leah's, long as both components are functioning. We're told that among 20 of these systems operating independently of each other were asked to find the probability that at least half of the morgue for more than three months Well, first of all, let's try to find the value of C. So we have. It's a probability distribution that one is equal to the integral from zero 10. This is an accident. Value the ex contagious and then the integral from 0 to 10 minus X is the maximum value that why could take for a given value of X of our function, which is C times 10 minus x plus y de y then DX and so we can pull out of C. We get C times after integrating. She takes a little bit of time, but it's not too difficult. 500 over three. Just integrate with respect, toe. Why first of them with respect to X And so it follows that C is equal to three over 500 which is equal to 0.6 Approximate. Now we have a probability p that one system lasts more than three months. Well, this is the same as the probability that both of its components last more than three months. This is because both finance are required for the system to work and to find a P well, P is the probability that X is greater than three. And at the component, why his lifetime? Greater than three. This is the same as p of three. Or I guess you could say F of three. Three integrated. So is a cumulative distribution function. So the integral from X could be a small is three in his greatest 10 and then integral from what could be a smallest three and as large as 10 minus x of f of X y. So we have that see we found was 0.6 times 10 minus x plus. Why de y DX? Once again, this integral is not impossible to solve. It's pretty easy to solve, however it takes time, so I'm going to skip the steps here will eventually get 0.64 approximately for P. Now, the number of systems out of 20 to meet the criteria follows a binomial distribution, in particular with n equals. 20 20 different systems and P Equalling are calculated value of 0.64 The probability that at least half of these 20 systems last more than three months. Well, this is some mhm. I'm going to find that at least half that's more than three months from K equals 10 2 20 of 20 choose K times 0.64 to the K. This is the probability of success and trial times the probability of failure, which is one minus 10.64 to the 20 minus K. It's a number of failures, so the reason we start with cake was 10 is because we have to include the number of success has to be at least 10 and this is going to be something. It's very difficult to compute by hand. So using computer, you find that this is approximately 0.0 000 117 incredibly small number, Very unlikely. That's over half of them are going to last this long.

Okay, so the question they have given us over here is the joint B D f. So I would just related the joint. Pdf. So then joined. PdF, which is given to us, is if off ex wife, which has given us X into eaters to minus X please to that is one place. Why in tow And this is for X rated an equal to zero wind. Why greater than equal to zero? And this is zero. Enter here. So we can say that the required probability, therefore, doesn't depend on divi values, So the required probability does not depend on the values. So because of this, to start with the first part, they have told us to find the probability off ex Graydon three. So there's there's no dependence on why so I can. Therefore, integrity, since they've given the limits for X, is greater than equal to zero and why is also greater than equal to zero. But for next, the limit is given as from greater than three, so we can take the limit off extra on three to infinity and the limit of violent means zero to infinity because there are no limits. Come in for a while except the one in the question. And then this is therefore fakes. And we have Dubai DX. So not to simplify this integration part. I'm wonder forced integrate with Why so I haven't dilation three to infinity as it is. Extra means as it is and I'm indicating forced. But by So I have these two minus x indu one plus y and you divide by coefficient of ice that is minus X okay. And the limiters from zero to infinity and the eggs as it is so we can cancel this X you're on. What remains is we can put the limits and that I'm taking minus outside the bracket limiters from three to infinity. And if you put the upper limit us to minus infinity Yes, zero minus. We're putting zero in place on fire. So this remains years to minus X with the ex. Finally what anger doesn't giggle? Three to infinity. It is to minus X the X. So far, they're doing the integration of attacks. We have eight days to minus x upon coefficient of excess minus one from three to infinity. So I'm thinking minus outside the black it again and If you simplify the whole expression using the calculator, you will have the answer. Approximately 0.5 So this is the approximate answer for the provider. Tee off X Greater than three. Not everyone to the next part in the Barbie. Their masters. Two questions. So I'm solving for the first part. So I am denoting that as b and the force part. So here I am saying that substituting the provided pdf the given bdf and integrate with respect Why talked in the marginal dear for X So to start with, we therefore find on the marginal pdf off X So the limits that are given to us I will force right down the formula. So we have effects off X. The individual is from minus infinity to infinity off f X y device. But I have a problem. The limits are from zero to infinity. So we take this to zero to infinity for function is exceeds two minus X in tow. One plus y Yeah, it's in the barber. One place, right. And this is Dubai. So now we're going to force to do the indignation with why So then, uh, we have X as it is an indignation will be easiest to minus X into one place. Right and coefficient of eyes minus x so limiters from zero to infinity. So we cancel X and then put the limits by taking minus outside. It is to minus infinity zero minus people zero in place off. Why? So you have CDs to minus X So the marginal pdf or fixes it is true minus X next just to find the model distribution off. Why? For this we're f y off. Why? Which will be in Diggle again? Minus infinity to infinity f off ex. Well, just given frozen the question integration will be with X. So our question is from zero to infinity and the function is exceeds two minus X into one plus. Why indicating with the ex? So now since we're invigorating with X so we will have to apply the integration You were here. So then I am going to integrate this. So this is excess. It is, and it is to integration is easiest to minus X in tow. One plus wife upon coefficient affects is minus off one place. Why, since we're indicating with X and indeed limiter zero to infinity. Then we have minus indigo zero to infinity. Dana Video Effexor Here is one and integration off Edie's to minus X in tow. One place by one coefficient of this is minus one plus five and the next in division has again with X. So then if I put the other limber, serious toe minus and 30 0 minus in place of exile putting zero So the complete upper limit as a list over number changes to zero. Then we changed the signs two plus and then we finally have a giggle zero to infinity leaders to minus sechs into one place while Windy X. So then if I just indicated this part, I have it is to minus X into one less Why one coefficient off this divided. And here we have a one place. Why already were here. So we shared this changes to the whole square and limit your responses to infinity. So then I have minus one upon one place. Why square outside the racket and I put the limits us to minus infinity zero minus It is Oh, what remains your is one because it is 20 is well so the final value for the Mahdi loopy dear for fires one upon one place. Why the whole square? So I can finally control down. So as marginal density functions, what for? X and y are for the expert. I'm writing it here. This can build in as it is to minus X for X there, Danny Contra zero and zero. Otherwise, right. And for if with why, we can write this as this is one upon one blessed by square. This is for violated any called a zero and zero on device. So we're done with the first part off me now moving on to the next part, which is the second part where they are asked to check if the very with our independent So we want to check, we're going to check if f off X comma y that we have been given This actually will do the individual marginal probability distributions of except by when multiplied. So we need to check if this is possible. So we say that since and wolf X comma y is X series two minus x into one place Why, right and FX off eggs into a fry off? Why is actually equal toe you days to minus X upon went beside the whole square for poor positive. So we therefore say that, uh, if Wolf X comma by in this case is not equal to the product of individual probability distributions and therefore we say then though through lifetime, really booze are No, no, not independent. Right? Okay. Not moving on to the next part of the part. See, they're asked to find the question where probability of X is greater than three. Or remember Tia Vias created and three. So in this case, when they say probably either X or Y agreed entries, we can take this as one minus probability off. Both extend by our lesson, equal a tree. So then this will change to one minus when they say less than equal to three. That means the limits will be from 0 to 3. So we put the double integral. Where would the limits as even three. And we did get the function so that X is doing a sex window. One place, right, the violent VX. So then I'm gonna now perform indignation. So I'm gonna force indignant with why. So we have e X as it is and used to minus X into one place. Why do it by coefficient of eyes minus X and limited, or a 0 to 3 with the x o X can cancel here and we have one plus what remains is indignities. You don't victory when you put the limits or here the function that getting is ah ee days to minus for X minus CDs to minus x VX. So next we're supposed to do this integration. So we have one plus installation of CDs to minus for Lex is ready by minus four here and then minus It is to minus X upon coefficient minus one. Limiters from zero to treat so far substituted a limited in one place. The upper limit treatment is what I get here is two minus trail upon minus fall plus E days two minus three and then lower limit banana boat. I get plus one by four minus once. If you simplify the holding The caliphate Oh, we get the answer approximately here as 0.3

You know this problem? We have been given the following joint probability distribution and we would like to find the conditional probability That X is between one and 2. Mhm. Given that why is equal to two, you can notice from this distribution that we can write our joint distribution as either the negative acts times either the negative Y. And as such, this tells us that they are in fact independent. So the probability that one is less than X is less than two, given that Y is to is just the same as the probability That access between one and 2 because they're independent. The conditional probability is relevant. And so this will be the integral from 1 to 2 of either the negative ads the X just looking at our marginal distribution friends, This is his negative. Either. The negative acts for max is 1-2. Yeah. Which is an idea of, you know the name of a second minus negative each of the negative first. And then if we want to get this as a decimal, just type this in on a catheter, This goes approximately .232 five


Similar Solved Questions

5 answers
CH13 Practice QuestionsName:Example 7: CzH1z0zNE;HKaLET KR1 FT-IA
CH13 Practice Questions Name: Example 7: CzH1z0z NE; HKaLET KR1 FT-IA...
4 answers
For the standard normal distribution, find:P(-2.68 <z <-2.17)
For the standard normal distribution, find: P(-2.68 <z <-2.17)...
5 answers
Problem 4 (10 points) A silicon p-n diode step junction maintained at room temperature is doped such that EF = Ev E/5 on the p-side and EF Ec KbT on the n-side. Take the temperature as 300K and the bandgap as 1.1 ev(a) Draw, to scale, the equilibrium energy band diagram for this junction: (b) Determine, from the diagram;, the built-in voltage Vbi:
Problem 4 (10 points) A silicon p-n diode step junction maintained at room temperature is doped such that EF = Ev E/5 on the p-side and EF Ec KbT on the n-side. Take the temperature as 300K and the bandgap as 1.1 ev (a) Draw, to scale, the equilibrium energy band diagram for this junction: (b) Deter...
5 answers
UG2SI3_6_U T
UG 2 SI 3_ 6_ U T...
5 answers
37 . Match the following compounds to the NMR spectra shown below: (8 points)NHzDEPT 135DEPT 90CHa200Ir = 1685PpMDEPT 135DEPT 90200120100IR = 1701DEPT 135DEPT 913C200 Ir = 1635DEPT 135DEPT 9018C200I=1718 13CPern
37 . Match the following compounds to the NMR spectra shown below: (8 points) NHz DEPT 135 DEPT 90 CHa 200 Ir = 1685 PpM DEPT 135 DEPT 90 200 120 100 IR = 1701 DEPT 135 DEPT 9 13C 200 Ir = 1635 DEPT 135 DEPT 90 18C 200 I=1718 13C Pern...
5 answers
Count to the number that is specified on the clock that is indicated.55 on a 12-hour clockThe number is
Count to the number that is specified on the clock that is indicated. 55 on a 12-hour clock The number is...
5 answers
Find the given limit:x2 _ 25 lim X-5 X-5Select the correct choice below and fill in the answer box within your choice_A. x2 _ 25 lim X-5 X-510] (Simplify your answer:)OB. The limit does not exist.
Find the given limit: x2 _ 25 lim X-5 X-5 Select the correct choice below and fill in the answer box within your choice_ A. x2 _ 25 lim X-5 X-5 10] (Simplify your answer:) OB. The limit does not exist....
5 answers
Evcrdttcn Nuntony ULvaik Wnna ailut Luan proteh ohydt Bhaping 126 + Mhoro thc {Ime In Yojn and arowh Nte during those vears | approtimated neight centimeters me setdlings a cenumoten whan plantodHow tAl are te #hnubs wlon tneyBoh}
evcrdttcn Nuntony ULvaik Wnna ailut Luan proteh ohydt Bhaping 126 + Mhoro thc {Ime In Yojn and arowh Nte during those vears | approtimated neight centimeters me setdlings a cenumoten whan plantod How tAl are te #hnubs wlon tney Boh}...
1 answers
For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. $$ \overrightarrow{P Q}-\overrightarrow{P R} $$
For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. $$ \overrightarrow{P Q}-\overrightarrow{P R} $$...
5 answers
IfA =[ %1-[8 Ic-E #-[4 &-G calculate (if possiblel) A + E A-1 B-1 ~3x 2y 2z Use Gauss-Jordan to solveFind the derivative of the following functions: Sx3 8x? + Tx + 15 52+81--7~+07 70--47+5 8x2 Tx + 15)(2x2 + x - 5) 8x? 7x + 15)100 Give that 25 = 32 use the tangent line of f(x) = xSto approximate2.0001510 Find the point of diminishing returns of the function _4x3 + .lx? + ,6x -
IfA = [ %1-[8 Ic-E #-[4 &-G calculate (if possiblel) A + E A-1 B-1 ~3x 2y 2z Use Gauss-Jordan to solve Find the derivative of the following functions: Sx3 8x? + Tx + 15 52+81--7~+07 70--47+5 8x2 Tx + 15)(2x2 + x - 5) 8x? 7x + 15)100 Give that 25 = 32 use the tangent line of f(x) = xSto approxima...
4 answers
2. You have a domain that is approximately 96 AA residues long: Knowing that domains within a protein structure are typically repeated, at least twice, what is the molecular weight (in kD) ofthree of these domains (Hint: assume each AA is 110 g mol-1). Show your work below to receive full credit
2. You have a domain that is approximately 96 AA residues long: Knowing that domains within a protein structure are typically repeated, at least twice, what is the molecular weight (in kD) ofthree of these domains (Hint: assume each AA is 110 g mol-1). Show your work below to receive full credit...
5 answers
Note that in the previous two questions we were interested only in the X-components of the velocity the Noomponents cepainZero aii the time Now the Same impulse i5 applied along the positive y-direction as shown below: Find the X- and y-components of the final velocity of the object:mThe X-component of the final velocity, V3xUnits Select an answerThe Y-component of the final velocity: VzyUnits Select an answerImp
Note that in the previous two questions we were interested only in the X-components of the velocity the Noomponents cepainZero aii the time Now the Same impulse i5 applied along the positive y-direction as shown below: Find the X- and y-components of the final velocity of the object: m The X-compon...
5 answers
Write an inequality for the graph below
Write an inequality for the graph below...
5 answers
Off these (HzO, BFz and CHa) whlch has a central atom that often does not obey the octet rule?
Off these (HzO, BFz and CHa) whlch has a central atom that often does not obey the octet rule?...
4 answers
Question 36 (4.5 points} Suppose retail organization wishes to compare' the average revenue generated by first-time customers when comparing two versions of their website: Version A or Version B_Each first-time customer was only shown only one version of the website (either version A or version B)Which method would the organization use to analyze this data?independent t-testpaired t-test
Question 36 (4.5 points} Suppose retail organization wishes to compare' the average revenue generated by first-time customers when comparing two versions of their website: Version A or Version B_ Each first-time customer was only shown only one version of the website (either version A or versio...
5 answers
Redo for 100%.You will necd Poisson tableQuestion 33 ptspage 178, #30 decimal placesAnswers to 4a) P(X-4)b) P(X<4)c) P(X<=4)Previous
Redo for 100%. You will necd Poisson table Question 3 3 pts page 178, #30 decimal places Answers to 4 a) P(X-4) b) P(X<4) c) P(X<=4) Previous...

-- 0.023161--