ENGR 323 BHW 15 Van Bonn 1/7
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1 ENGR 33 BHW 5 Van Bonn /7 4.4 In Eriss and 3 as wll as man othr situations on has th PDF o X and wishs th PDF o Yh. Assum that h is an invrtibl untion so that h an b solvd or to ild. Thn it an b shown that th PDF o Y is [ ] a I X has a uniorm distribution with A and B driv th PDF o Y ln X b Wor ris usin this rsult. Wor Eris 3b usin this rsult. Baround Inormation: Thr ar our stps to dtrmin th PDF o Y.. Find. Find ' 3. Find th ran o or 4. Us Equation and plu in and ' and solv or th PDF Not: Y h must b invrtibl to produ X and ' must ist. SOLUTION a For part a w ar told that X is uniorml distributd [X~UniormAB]. Our tt p. 7 dins th PDF o a ontinuous uniorm random variabl as A B B A othrwis B - A A B
2 ENGR 33 BHW 5 Van Bonn /7 Thror or 4 part a th PDF o th random variabl X is othrwis Furthrmor th problm statmnt dins th ollowin h Y ln X STEP Usin albrai manipulation w solv quation or as a untion o as ollows Y ln X 3 STEP Finall solvin or th drivativ o w hav STEP 3 [ ] 4 Sin thn an valu o on th intrval rom A to B Whih raiss th qustion: What ar th intrval valus o Y? From th PDF o X w now that dins th ran o valus o. Thror to din th ran o valus that th random variabl Y an bom w solv th ollowin or B plottin variabl Y. I thn and thror < Fiur No. an b usd to raphiall dtrmin th ran or th random
3 ENGR 33 BHW 5 Van Bonn 3/7 p- Ran o Y + "Asmptot" Fiur No. Th raph o usd to dtrmin th ran o. STEP 4 Pluin quations 3 and 4 into quation rsults in th ollowin PDF or th random variabl Y [ ] [ ] othrwis Fiur No. is a raphial rprsntation o th PDF o th random variabl Y. p- Ran o Y Not: As approahs ininit th ara undr th urv o p- is on as ptd Fiur No. Th PDF o Y CONCLUSION: Th distribution o th nativ o th natural lo o a uniorm distribution is an ponntiall distributd random variabl.
4 ENGR 33 BHW 5 Van Bonn 4/7 4.4 b Wor ris usin this rsult. Lt Z hav a standard normal distribution and din a nw random variabl Y Z + µ. Show that Y has a normal distribution with paramtrs µ and. Solution b Qustion dins th random variabl Z to hav a standard normal distribution [Z~Normµ ]. From our tt p. 79 us th standard normal random variabl Z and plu it into th PDF o th normal random variabl X as dind on p. 7 o our tt to produ th PDF o Z. Eq p.79: Z µ X X µ Z Eq p.75: µ π To produ th PDF o th standard normal distribution o our random variabl Z substitut Equation i into Equation and rpla with. z z - z π < z < IF... and th problm tlls us... STEP thn... as w ptd. hz Y Z + µ µ 5 STEP B solvin or th drivativ o w hav 6
5 ENGR 33 BHW 5 Van Bonn 5/7 STEP 3 I and larl < Y Z < Z + µ < Y < whr and µ ar onstant STEP 4 B substitutin Equation 5 and 6 into Equation rom part a w produ th ollowin µ µ π π µ - < < Fiur No.3 is th PDF or an normal distribution with paramtrs µ and. I µ and thn this PDF boms th standard normal distribution n n standard dviations rom th man Fiur No.3 PDF or a normal distribution with n standard dviations rom th man µ. Noti that i and th µ thn this is th standard normal distribution
6 ENGR 33 BHW 5 Van Bonn 6/7 4.4 Wor ris 3 b usin this rsult. 3 b I X has a amma distribution with paramtrs and what is th probabilit distribution o Y X? Solution First w ar told ~ GAMMA X From prvious handouts w now th PDF o X > othrwis 3 b tlls us that Y X STEP suh that w an dtrmin STEP and solvin or th drivativ o w hav STEP 4 Pluin Equations 7 and 8 into Equation rom part a w hav th ollowin ] [ Combin ths two and ou t
7 ENGR 33 BHW 5 Van Bonn 7/7 And i this is rwrittn as suh... It is as to s that Y has a amma distribution with th onl dirn rom X bin th onstant tims. Thror w an sa ~ GAMMA Y and has th ollowin PDF > othrwis STEP 3 W now this to b tru baus an random variabl with a amma distribution is ratr than zro and Y~Gamma
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