Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12
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1 Enginring Bautiful HW #1 Pag 1 of Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th first componnt xcds? b. What ar th marginal pdf s of X and Y? Ar th two liftims indpndnt? Explain. c. What is th probability that th liftim of at last on componnt xcds? f(x, Componnt X Liftim 7 Componnt Y Liftim Figur 1. Minicomputr Componnt Liftim Joint Probability Dnsity Function Hr ar som usful trms and quations that will hlp us to solv this problm. Dfinitions and Equations Joint Probability Dnsity Function Lt X and Y b continous rv s. Thn is th joint probability dnsity function for X and Y if for any two-dimnsional st A P [( X, Y) A] dx dy = A In particular, if A is th two-dimnsional rctangl {( x, : a x b, c y d}, thn
2 Enginring Bautiful HW #1 Pag of 6 P [( X, Y) A] = P( a X b, c Y d) dy dx b = a c d Marginal Probability Dnsity Function Th marginal probability dnsity functions of X and Y, dnotd by f x (x) and f y (, rspctivly, ar givn by f x ( x) = dy, for < x < f y ( = dx, for < y < a. What is th probability that th liftim X of th first componnt xcds? Th probability that X> is found using th dfinition of th Joint Probability Dnsity Function. This probability dos not dpnd on th y valus. You hav to considr all possibl valus of y, so you must intgrat ovr all valus of y as wll as valus of x from to. You might want to gt out your calculus book to hlp you with th following intgrals. P (1+ ( X > ) = dy dx First you intgrat with rspct to y from through. Notic that w ar first intgrating only th insid of th doubl intgral. (1+ dy dx Th rsulting intgral is a function of x, ( 1+ y y dy = = x = ax ax * Hlpful Hint: Rmmbr that dx = a
3 Enginring Bautiful HW #1 Pag of 6 You thn hav to intgrat this with rspct to x from to, obtaining dx = =.5 This mans that thr is about a 5% chanc that th liftim of componnt X xcds. b. What ar th marginal pdf's of X and Y? Ar th two liftims indpndnt? Explain. Th marginal probability dnsity functions of X and Y, dnotd by f x (x) and f y (, rspctivly ar givn by f x ( x) = dy, for < x < f y ( = dx, for < y < In ordr to comput th marginal dnsity function of X, w must intgrat th function with rspct to y. W alrady did this intgration in part a so you can rfr to that for th answr. dy = ( 1+ Th marginal pdf of X is x f x( x) = othrwis
4 Enginring Bautiful HW #1 Pag 4 of px(x) Liftim of Componnt X Figur. Marginal Probability Dnsity Function of X W must now intgrat th function with rspct to x in ordr to find th marginal dnsity function of Y. Intgration by parts is rquird hr. udv = uv vdu u = x dv = (1+ du = dx (1+ y ) v = (1 + (1+ (1+ x(1+ x(1+ dx = dx = ( 1+ (1 + (1 + (1 + x( = = (1 y ) + (1 + * Hlpful Hint Bth said that w will not hav to intgrat by parts on th xam Now that w got through that, w know that th marginal pdf of Y is 1 f y ( = (1 + y othrwis
5 Enginring Bautiful HW #1 Pag 5 of py( Liftim of Componnt Y Figur. Marginal Probability Dnsity Function of Y Th qustion now asks us to dtrmin if th two liftims ar indpndnt. Th dfinition of th indpndnc for two rv's is = f ( x) f (, for all X and Y whn X and Y ar continuous (p. ) x y By computing this product w find that (1 + 1 (1+ It is clar that f(x, is not th product of th marginal pdf's. Thrfor, X and Y ar not indpndnt. c. What is th probability that th liftim of at last on componnt xcds? Th ky to solving this problm is to rmmbr that th kywords at last usually indicat that th asist mthod is to us th complimnt. Th complimnt is both X and Y ar lss than. P( X > Y > X > Y > ) = 1 P( X Y ) = (1+ y 1 dy dx = 1 dy dx
6 Enginring Bautiful HW #1 Pag 6 of 6 x 1 = 1 (1 ) dx = =. This mans that thr is a % chanc that at last on of th componnts liftims will xcd.
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