Two components of a computer have the complying with joint pdf because that their valuable lifetimes $X$ and also $Y$$f(x, y)=left{eginarrayccx e^-x(1+y) & x geq 0 ext and y geq 0 \ 0 & ext otherwise endarray ight.$(a) What is the probability the the lifetime $X$ of the an initial component over 3$? ?$(b) What are the marginal pdfs the $X$ and also $Y ?$ room the 2 lifetimes independent? Explain.(c) What is the probability the the lifetime of at least one ingredient exceeds 3??

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Probability with Applications in Engineering, Science, and technology (precalculus, calculus, Statistics)

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Okay, so the inquiry they have provided us over right here is the joint B D f. So ns would simply related the joint. Pdf. So then joined. PdF, which is provided to us, is if turn off ex wife, i beg your pardon has given us X right into eaters to minus X please to that is one place. Why in tow and this is for X rated an same to zero wind. Why higher than equal to zero? and this is zero. Get in here. For this reason we can say the the compelled probability, therefore, doesn't depend on divi values, therefore the required probability does not depend on the values. So because of this, to begin with the first part, they have actually told us to discover the probability off ex Graydon three. So there's there's no dependency on why so ns can. Therefore, integrity, since they've given the borders for X, is better than same to zero and also why is likewise greater 보다 equal come zero. Yet for next, the border is given as from higher than three, for this reason we deserve to take the limit off extra on 3 to infinity and the border of violent method zero to infinity because there space no limits. Come in because that a while other than the one in the question. And also then this is because of this fakes. And also we have actually Dubai DX. So not to leveling this integration part. I'm wonder forced incorporate with Why so i haven't dilation three to infinity together it is. Extra way as it is and I'm describe forced. However by therefore I have actually these two minus x indu one to add y and you divide by coefficient of ice that is minus X okay. And the limiters from zero to infinity and the eggs together it is so we have the right to cancel this X you're on. What stays is we have the right to put the limits and also that I'm acquisition minus outside the clip limiters from three to infinity. And if you placed the upper limit us to minus infinity Yes, zero minus. We're putting zero in place on fire. Therefore this stays years come minus X v the ex. Lastly what anger doesn't giggle? three to infinity. The is to minus X the X. For this reason far, they're law the integration that attacks. We have eight days come minus x upon coefficient of overabundance minus one from three to infinity. So I'm thinking minus outside the black color it again and also If you simplify the entirety expression using the calculator, friend will have actually the answer. About 0.5 therefore this is the approximate answer for the provider. Tee off X higher than three. Not everyone come the next part in the Barbie. Their masters. 2 questions. So I'm fixing for the first part. So ns am denoting that as b and the force part. So below I am saying that substituting the detailed pdf the given bdf and integrate through respect Why talked in the marginal dear because that X for this reason to start with, we therefore find top top the marginal pdf turn off X for this reason the boundaries that are given to us I will force right down the formula. So us have impacts off X. The individual is indigenous minus infinity to infinity off f X y device. However I have actually a problem. The borders are from zero come infinity. So us take this to zero to infinity for role is exceeds two minus X in tow. One to add y Yeah, it's in the barber. One place, right. And this is Dubai. So currently we're walk to pressure to carry out the indignation through why so then, uh, we have actually X together it is an indignation will certainly be easiest to minus X into one place. Right and coefficient of eye minus x therefore limiters indigenous zero to infinity. So us cancel X and also then placed the borders by acquisition minus outside. The is come minus infinity zero minus world zero in ar off. Why? for this reason you have actually CDs to minus X for this reason the marginal pdf or fixes that is true minus X next simply to find the model distribution off. Why? for this we're f y off. Why? Which will certainly be in Diggle again? Minus infinity to infinity f turn off ex. Well, just given frozen the inquiry integration will be through X. Therefore our inquiry is indigenous zero come infinity and the duty is exceeds two minus X right into one plus. Why indicating v the ex? so now because we're invigorating through X so we will have to apply the integration You to be here. For this reason then i am walk to incorporate this. For this reason this is excess. It is, and also it is come integration is most basic to minus X in tow. One plus wife upon coefficient affect is minus off one place. Why, since we're indicating with X and indeed limiter zero to infinity. Climate we have actually minus indigo zero to infinity. Dana video Effexor below is one and also integration off Edie's come minus X in tow. One ar by one coefficient of this is minus one to add five and also the next in division has again through X. So climate if I placed the other limber, serious toe minus and also 30 0 minus in place of exile putting zero for this reason the complete upper limit together a list over number changes to zero. Then we changed the indications two plus and then we lastly have a giggle zero to infinity leaders to minus sechs right into one place while public X. So climate if ns just suggested this part, I have it is to minus X right into one much less Why one coefficient turn off this divided. And also here we have actually a one place. Why already were here. For this reason we shared this changes to the whole square and also limit her responses to infinity. So then I have actually minus one top top one place. Why square outside the racket and I put the borders us to minus infinity zero minus it is Oh, what continues to be your is one due to the fact that it is 20 is well so the final value for the Mahdi loopy dear because that fires one top top one place. Why the entirety square? therefore I can finally regulate down. So together marginal thickness functions, what for? X and also y are for the expert. I'm composing it here. This can construct in together it is come minus X for X there, Danny Contra zero and also zero. Otherwise, right. And also for if v why, we have the right to write this together this is one upon one blessed through square. This is for violated any type of called a zero and also zero on device. Therefore we're done v the very first part off me now moving on to the next part, i m sorry is the second component where they are asked to examine if the very with our independent therefore we desire to check, we're walking to examine if f turn off X comma y the we have actually been offered This actually will execute the individual marginal probability distribution of except by as soon as multiplied. So we need to examine if this is possible. So we say the since and wolf X comma y is X series two minus x right into one place Why, right and FX off eggs right into a fried food off? Why is actually equal toe friend days to minus X ~ above went alongside the entirety square for poor positive. So we as such say that, uh, if wolf X comma by in this situation is not equal to the product of individual probability distributions and also therefore us say then though with lifetime, yes, really booze room No, no, no independent. Right? Okay. Not relocating on come the next component of the part. See, they're asked to discover the concern where probability that X is better than three. Or remember Tia Vias created and three. So in this case, when they say probably either X or Y i agree entries, we have the right to take this as one minus probability off. Both expand by our lesson, same a tree. So climate this will adjust to one minus as soon as they say much less than same to three. That method the boundaries will be from 0 to 3. Therefore we placed the dual integral. Whereby would the boundaries as even three. And also we did obtain the duty so that X is law a sex window. One place, right, the violent VX. So climate I'm gonna now perform indignation. So I'm gonna pressure indignant with why. For this reason we have actually e X as it is and also used come minus X into one place. Why carry out it by coefficient of eyes minus X and also limited, or a 0 come 3 v the x o X can cancel here and also we have actually one plus what stays is indignities. Friend don't victory once you placed the borders or right here the function that acquiring is ah ee days come minus because that X minus CDs to minus x VX. So following we're supposed to perform this integration. Therefore we have one add to installation that CDs to minus for Lex is all set by minus 4 here and also then minus it is come minus X ~ above coefficient minus one. Limiters native zero come treat so much substituted a restricted in one place. The top limit treatment is what I acquire here is two minus trail upon minus autumn plus E days two minus three and also then lower limit banana boat.

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I get plus one by four minus once. If you simplify the hold The caliphate Oh, we gain the answer around here as 0.3