4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b

Transcription

1 4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs movi, is uniformly disribud bwn 00 and 000. a Wha ar h man and varianc? b Wha is h probabiliy ha h unblivabiliy lvl is abov 500? 4. Exponnial Disribuion Df n: An xponnial random variabl rprsns h lngh of an inrval from a crain poin unil h nx succss in a oisson procss. As such, h xponnial disribuion is lik h gomric by looking for h prob. of h nx succss. A coninuous random variabl has an xponnial disribuion wih paramr avg. # of succsss pr uni inrval whn is pdf and cdf ar x f x x and x, x 0, 0 Also, µ E and V Lik h oisson disribuion, us consisn unis o dfin an xponnial random variabl and paramr. Also, an xponnial curv is skwd o h righ, dcrasing xponnially; as incrass, h skwnss of h curv incrass. S pag 89. Th xponnial and oisson disribuions ar inrchangabl. L an xponnial random variabl wih paramr dno h im unil h nx succss. L N dno h # of succsss during a spcific im inrval x i.. N is a oisson random variabl wih paramr x. Thn, h following rlaionship xiss bwn and N: 0 x x x N 0 -x 0! Thrfor, x x x -x. By diffrniaing x, pdf of is d x x f x, x 0 dx Thus, w hav an xponnial disribuion.

2 oisson Exponnial Lngh of im inrval Consan Variabl # of succsss Variabl N Consan As sn in 3.., h lack of mmory propry allows for xclusion of h hisory of prvious oucoms such ha h im can b rs. In ohr words, roof + + and Thus,. Only h xponnial disribuion has his propry among coninuous disribuions as did gomric among h discr disribuions. This propry dos imply, hough, ha h valu of dos no chang wih, i MUST b consisn. Ex4. rsum a oisson procss linking vry singl gam a am plays so ha h im is, ssnially, coninuous. Th # of goals by on am has a man of.75 pr gam or, 60 minus. a Drmin h pdf of h im ; uni: 5 min unil h nx goal. b ind h probabiliy ha hr ar no goals for a las 5 minus by using boh h xponnial and oisson disribuions.

3 c ind h man and sandard dviaion of. 5. Normal Disribuion Df n: A c.r.v. has a normal disribuion wih man µ and varianc whn is pdf is xµ f x - x π Th paramrs µ rangs from - o and 0 ar qual o E and V, rspcivly. A normal disribuion, dnod as Nµ,, is symmric abou µ and bllshapd. Th symmry of a normal curv implis µ µ 0.5. Th paramrs affc h cnr and shap of h curv. diagrams drawn in class show ach ffc robabiliy of h Normal Disribuion a.k.a. h Empirical Rul: Th ara byond ±3 is qui small lss han 0.0. As in igur 5.6, h wo paramrs combin for noworhy propris:. 68% of obsrvaions li wihin of µ.. 95% of obsrvaions li wihin of µ % of obsrvaions li wihin 3 of µ. This las propry implis ha 6 is approximaly h full widh of a normal disribuion. Df n: A normal r.v. Z has a sandard normal disribuion whn µ 0 and. Th cdf of Z is dnod by Φz Z z Tips & ricks: - diagrams ar hlpful - Complmn: Z z Z z Z z - Symmry: Z z Z z - a Z b Z b Z a - If z 0, hn z Z z Z z

4 Ex5. Exampls wih z-scors finding prob.: a Z -3.4 b Z.44 Z.44 OR Z.44 Z.44 c -3.4 Z.44 Z.44 Z -3.4 d -.00 Z.00 Z Sandardizing a Normal Disribuion: ~ Nµ, µ and Z ~ N0,. Wha is Z? Z, z µ x µ x Z z x µ Ex5. Exampls wih z-scors sandardizing: ind h following probabiliis for ~ N75, 6.5 : a Wha is h probabiliy of ging a valu grar han 94.5? b Wha is h probabiliy of ging a valu bwn 7.75 and 84? Idnifying valus: Using h ara undr h curv, you can find appropria z valus; so, wha ar h corrsponding x valus? x µ + z Ex5.3 Exampls wih z-scors finding valus: Us h sam as in Ex5. o answr h following: a Wha valu dnos h op 5%?

5 b Wha valus bound h middl 70% of h daa? 5. rvisis.6 for normal random variabls 5.3. Normal Approximaion o Binomial Whn p is clos o 0 or, h binomial disribuion is qui skwd. In hs cass, normal approximaion is inappropria. If, howvr, np 5 AND n p 5, hn h binomial disribuion, Bn, p, is approximaly normal wih µ np and np p. Thus, if ~ Bn, p, hn x np x Φ np p x 0.5 np x Φ np p Nos:. Th 0.5 rprsns a coninuiy corrcion o improv h approximaion.. Th xbook liks h condiion valu of 5, bu 5 is br and coincids br wih lar marial, so xams will prfrably avoid valus bwn 5 and 5. Ex5.4 Suppos ha ~ B00, 0.. ind 5 from a h binomial disribuion and b if h approximaion is appropria, h approxima normal disribuion. No also h following calculaions: Binomial Normal approximaion using using using 5 0.5