Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl spacs. In this chaptr, w introduc som of th common PDFs for continuous sampl spacs. Th goal of this sction is to bcom familiar with ths probabilit distributions and, whn givn a word problm, know which PDF is appropriat. Continuous Uniform Distribution (p. 43) Th continuous uniform distribution dscribs a sampl spac whr vr point within th spac is quall (uniforml7) likl. Th dnsit function of th continuous uniform random variabl X on th intrval [A,B] is f(;a,b) B A for A B othrwis (6.) Th man is (from quation (7.)) µ A + B (6.) and th varianc is (from quation (7.4)) ( B A) (6.3) Eampl 6.: Th dfault random numbr gnrators for most programming languags provids a uniform distribution on th intrval [,]. What is th probabilit that a FORTRAN random numbr gnrator ilds (a) X.5 (b) X<.5, (c ).5, (d) X>.5, ().<X<.5? (a) Th probabilit of gttting actl an numbr in a continuous sampl spac is zro. (b) B quation (4.3), th dfinition of a contiuous PDF:.5.5 P(X <.5) P( < X <.5) f()d d.5 (c)
P(X.5) P(X <.5) + P(X.5) P(X <.5).5 (d) P(X >.5) P(X <.5) -.5.75 () P(. < X <.5).5.5 f()d d...5 Normal Distribution (p. 45) Th normal distribution is th most important continuous PDF in statistics and in scinc. It is also known as th Gaussian distribution. It s shap has givn ris to th nicknam th bll curv. Th dnsit function of th normal random variabl X, with man µ and varianc, is normal distribution f(; µ, ) π µ (6.4) You will notic immdiatl that this distribution is diffrnt than th PDF s w v studid prviousl. Th normal PDF is dfind b th man and varianc, µ and. In our prvious cass, th PDF dfind µ and. Th ffct of th man on th normal PDF is shown blow..5.4 m m4 m8 f().3... -3.. 7. s Th ffct of th varianc on th normal PDF is shown blow
.5.4 s f().3.. s.5 s3. -5. -3. -.. 3. 5. m Som charactristics of th Normal PDF: Th mod, (th most probabl valu occurs at th man). Th curv is smmtric about th man. Th curv has inflction point at µ ±. (Rmmbr: inflction point ar whr th scond drivativ is zro, whr th curv changs from concav up to concav down. Th normal curv approachs th -ais as w mov from th man. Th toral ara undr th curv and abov th -ais is on (as it is for all PDF s.) b Th probabilit P(a < X < b) f()d is th ara undr th normal curv a btwn a and b. Th Normal PDF with µ and is calld th Standard Normal PDF. An Normal PDF, f(; µ, ),can b convrtd to a standard normal PDF, f (z;, ), µ with th chang of variabl z Th Normal function in quation (6.4) cannot b analticall valuatd. It can ithr b numricall valuatd or ou can convrt our normal PDF to th Standard Normal PDF and us th tabulatd valus in Tabl A.3 of WMM. Th valus in Tabl A.3 ar listd b z-valu and man P ( < X < z), th probabilit that a valu is lss than z. Th Normal PDF f(; µ, ) is qual to th binomial PDF n b(;n,p) p q n with µ np and npqwhn n 3
Eampl 6.: (ampl 6.7 WMM, p. 53) A tp of battr lasts on th avrag 3. ars with a standard dviation of.5 ars. What is th probabilit that th battr will last (a) lss than.3 ars? (b) btwn.5 and 3.5 ars? (c) mor than fiv ars? (a) First stp: convrt th non-standard Normal PDF to th standard Normal PDF application of normal distribution z. 3 µ.3 3..5.4 P (X.3) P(Z.4).88 That last numbr coms from Tabl A.3 of WMM. (b) z. 5 µ.5 3. µ 3.5 3.., z 3. 5..5.5 P(.5 X (c) 3.5) P(. Z.) P(Z.).843 -.587.686 P(Z.) z. µ 4. 3..5 4. P(4. X) P(. Z) P(. Z).977.8 Gamma and Eponntial Distribution (WMM, p. 66) Two othr tps of PDF s commonl usd in nginring ar th Gamma and Eponntial Functions. Th gamma function is th basis of on th Gamma Distribution. Th Gamma Function has th following dfinition (Dfinition 6., p. 67) Γ( α) α d for α > (6.5) Th Gamma function has som spcial valus: Γ ( ), Γ ( n) (n )! whr n is a positiv intgr, and Γ ( / ) π. Th Gamma distribution is dfind for th continuous random variabl X with paramtrs α and β as 4
f Γ (; α, β) β α Γ( α) α- -/ β for > lswhr (6.6) Whn a, th Gamma distribution is calld th ponntial distribution f (; β) β -/ β for > lswhr (6.7) Th man and varianc of th Gamma distribution ar µ αβ and αβ (6.8) Eampl 6.3: Th liftim of sparkplugs is masur in tim, t, and is modlld b th ponntial distribution with an avrag tim to failur of 5 ars, β 5. If nw sparkplugs ar installd in an 8-clindr ngin and nvr rplacd, what is th probabilit that m spark plugs ar still aliv at th nd of t i ars whr m 8 and t i,.5, 5, 7.5,. Th sparkplugs ar indpndnt of ach othr. Th probabilit that a singl indpndnt spark plug is still aliv at th nd of t i ars is givn b: P(t i < t) f(t; β)dt β ti ti -t/ β dt -ti/ β W can comput this for an dsird valu of t i. Now, w nd that probabilit that m of n8 sparkplugs ar still functioning at t i, givn th probabilit abov. This is prcisl th function of th binomial distribution, b(;n,p) whr m, th numbr of functioning sparkplugs, n8, th numbr of total sparkplugs; and p P(t i < t), th probabilit that a singl sparkplug maks it to tim t i. W can calculat b(m,8, P(t i < t) ) for all valus of m, (naml m 8) and for svral valus of t i. This is don in th graph blow. Lt s chck that plot out. Th plot shows th probabilit that m spark plugs function at tim t. At an tim t, th sum of th probabilitis is. At tim nar, it is most probabl that all 8 sparkplugs still function. At si ars, it is most probabl that onl sparkplugs still function, followd b 3,, 4,, 5, 6, 7, 8. At twlv ars, it is most probabl that no sparkplugs function anmor. 5
Probabilit that m parts function.5.4.3.... 3. 6. 9.. t (ars) m 3 4 5 6 7 8 Chi-Squard Distribution (WMM, p. 7) An additional spcial cas of th Gamma Distribution is obtaind whn αv/ and β, whr v is calld th dgrs of frdom and is a positiv intgr. Th random variabl X has a Chi-Squard Distribution with v dgrs of frdom if f χ (;v) v / Γ(v / ) v/- -/ for > lswhr (6.9) Th man and varianc of th Chi-squard distribution ar µ v and v (6.) W will ncountr applications of th Chi-squard distribution in th following sction. In particular, th Chi-squard distribution is important componnt of statistical hpothsis tsting and stimation. Functions of Random Variabls An PDF of a variabl can b transformd into a PDF of a function of that variabl. In gnral, f () can b transformd to g(()). Eampl 6.4: Whn w changd th variabl of a normal distribution, f(; µ, ), to th standard normal distribution, f (z; µ, ), w usd th transformation, 6
µ z() Eampl 6.5: Lt X b a continuous random variabl with PDF f() for < < 5 lswhr thn th probabilit, P(<X) is givn b P( > X) f( )d d 4 Find th probabilit distribution for, whr () +. Rarranging for ilds (-)/ and dd/ and g()f()f((-)/) with limits ()4 and (5). So, g() f(( and )/ ) ( )/ P( > X) P(() > Y(X)) 4 f( )d d g( ) 4 d 4 P(() > Y(X)) 4 4 96 96 4 4 96 Blow ar two plots which show f() and F()P(>X), th cumulativ PDF, as a function of. And g() and G()P(>Y), th cumulativ PDF, as a function of. 7
f() and F()...8.6.4. f() F().... 3. 4. 5. 6.. g() and G()..8.6.4. g() G().. 4. 6. 8... 4. 8