ENGI 4421 Discrete Probability Distributions Page Discrete Probability Distributions [Navidi sections ; Devore sections
|
|
- Cuthbert Jenkins
- 5 years ago
- Views:
Transcription
1 ENGI 441 Discrete Probability Distributios Page 9-01 Discrete Probability Distributios [Navidi sectios ; Devore sectios ] Chater 5 itroduced the cocet of robability mass fuctios for discrete radom quatities. The oly stadard distributio itroduced the is the discrete uiform distributio, whose.m.f. is 1 1,,, [ad zero otherwise] This chater will itroduce some more stadard robability mass fuctios. Beroulli Probability Distributio Where the outcome of a sigle trial ca take oe of oly two ossible values, success or failure, the Beroulli radom quatity mas success to X = 1 ad failure to X = 0. Success ad failure are a comlemetary air of simle evets. Success ca mea heads o oe fli of a coi, or defective whe a sigle item is tested, or total is 7 o oe roll of two dice, etc. If the robability of a success is, the the.m.f. follows immediately: The mea ad variace are calculated easily: 1 X E 0 1 ad E i 0 1 i X otherwise X X X V E E 1
2 ENGI 441 Discrete Probability Distributios Page 9-0 Eamle 9.01: Let X = 1 corresod to total score o oe roll of a air of fair stadard si-sided dice is 7 ad X = 0 i the evet of some other total, the X has a Beroulli distributio. Fid the arameter of this distributio ad fid the mea ad variace of X. There are 36 equally likely ossible airs of scores. Si of them roduce a total of 7. Therefore EX ad VX The Biomial Distributio The most imortat discrete robability distributio is the biomial distributio. develo the robability mass fuctio through a eamle. Here we Eamle 9.0 It is kow that each comoet, as it emerges from a eerimetal roductio rocess, has a robability of 0% to be defective (ad therefore a robability of 80% to be good). A radom samle of te comoets is draw. (a) Fid the robability that eactly three comoets i the radom samle are defective. I a sigle trial (a sigle comoet i the radom samle) let S = success (the comoet is defective) ad F = failure (the comoet is good) ( = S ~ ) [It may seem strage to associate success with defective, but it is the umber of defective items i which we are iterested.] This is a Beroulli trial (a air of comlemetary outcomes oly), with costat P[S]. Also let E = the desired evet (three defective ad seve good comoets i the radom samle) The oe way i which the desired evet ca occur is if the first three comoets tested are all defective ad the remaiig seve are all good. The robability of this outcome (SSSFFFFFFF i that order) is S F P
3 ENGI 441 Discrete Probability Distributios Page 9-03 Eamle 9.0(a) (cotiued) However, iside the evet E, the three successes may occur aywhere amog the te trials. The umber of distict rearragemets of three successes i te trials is 10 C Therefore P[E] = (b) Fid the robability mass fuctio for X = (the umber of defective comoets i the radom samle). P 10 S F But these successes ca be rearraged amog the 10 trials i 10 C distict ways. Therefore the robability mass fuctio of X is P[eactly successes] = P[X = ] =.0.80 C ( = 0, 1,,..., 10) We ca geeralize this result further. Let the roortio of defective items i the oulatio be (rather tha 0%) ad let the samle size be (rather tha 10). The the biomial robability mass fuctio becomes P ;, 1 X b I this eamle, we have draw a radom samle from a cocetual oulatio, so that every trial is ideedet of every other trial.
4 ENGI 441 Discrete Probability Distributios Page 9-04 Coditios for a discrete radom quatity to have a biomial.m.f.: I geeral, if the radom quatity X reresets the umber of successes i trials the X will have a biomial robability distributio if ad oly if the followig coditios all hold: (1) Each trial has eactly two comlemetary outcomes ( success ad failure ); () The [ucoditioal] robability of success is costat across all trials; (3) The outcome of each trial is ideedet of all other trials; (4) The samle size is fied. Coditio (3) holds oly if the samlig is either with relacemet or from a ifiite oulatio (or both). If the samlig is without relacemet from a fiite but large oulatio, the coditio (3) may be aroimately true. Eamle 9.03 Show that coditio (3) is ot satisfied if a radom samle of size is take from a oulatio of size 5 with 40% successes. Show that coditio (3) is almost satisfied if a radom samle of size is take from a oulatio of size 5000 with 40% successes. S S F F F Let S 1 = success o trial 1 ad S = success o trial the P S but P S S Therefore the outcomes of the two trials are ot ideedet. [The eact robability distributio is hyergeometric.],000 S s 3,000 F s Let S 1 = success o trial 1 ad S = success o trial 000 the P S but P S S1 1, ,999 Therefore the outcomes of the two trials are early ideedet ad the biomial.m.f. may be used as a ecellet aroimatio to the true robability distributio.
5 ENGI 441 Discrete Probability Distributios Page 9-05 The biomial cumulative distributio fuctio is y B ;, b ;, y 0 = b(0;, ) + b(1;, ) + b(;, ) b(;, ) The biomial cdf B(;, ) is tabulated i Navidi (ad Devore), table A.1, for u to 0 ad values of that are iteger multiles of.05. It ca be evaluated for ay valid choice of (, ) usig the file at " Eamle 9.04 The robability mass fuctio of the radom quatity X is kow to be biomial with arameters = 10 ad =.0. Fid P[1 < X < 5]. P[1 < X < 5] = P[1 < X 4] = B(4; 10,.0) B(1; 10,.0) = (usig Table A.1) =.591 or, usig the Web file "Biomial.ls", B(4; 10,.60) B(1; 10,.60) = OR P[1 < X < 5] = P[X = ] + P[X = 3] + P[X = 4] C C3 C4 = = Recall that P a X b F b F a F where the c.d.f. is
6 ENGI 441 Discrete Probability Distributios Page 9-06 Eamle 9.04 (cotiued) A sreadsheet ca be used to carry out these calculatios. MINITAB ad Ecel both cotai the biomial mf ad cdf, as does the Ecel file at " To fid the values of B(4; 10,.0) ad B(1; 10,.0), the followig artial table of values for b(; 10,.0) ad B(; 10,.0), (draw from the Ecel file), may be used: = 10 = 0..m.f. c.d.f. P[X = ] P[X ] If a questio ivolvig the calculatio of several adjacet biomial robability masses arises i a test or eamiatio, the the aroriate table of values, similar to that show above, will be rovided with the questio aer. [Oe could also just add u the values of the.m.f. for, 3, 4 to fid P1 X 5 ]
7 ENGI 441 Discrete Probability Distributios Page 9-07 Eamle 9.05 Te er cet of all items i a large roductio ru are kow to be defective. A radom samle of 0 items is draw. (a) (b) (c) Prove that the radom quatity X = (the umber of defective items i the radom samle) has a biomial robability mass fuctio. Fid the robability that more tha two items i the radom samle are defective. How may defective items does oe eect to fid i the radom samle? (a) Let X = the umber of defective items i the radom samle. each trial (item) has a comlemetary air of outcomes (defective = success, good = failure ) P[success] = costat = 10% Trials are ideedet to a good aroimatio (because the radom samle is draw from a large oulatio) = 0 is fied. All four coditios are satisfied. Therefore X does follow a biomial robability distributio (at least to a ecellet aroimatio). P[X = ] = b(; 0,.10). (b) P[X > ] = 1 P[X ] = 1 B(; 0,.10) = OR (by direct calculatio of the.m.f. values), 1 P[X ] = 1 (P[X = 0] + P[X = 1] + P[X = ]) = 1 { (.9) 0 + 0(.9) 19 (.1) (.9) 18 (.1) } = 1 ( ) Side ote: P[X > ] = P[ < X 0] = B(0; 0,.1) B(; 0,.1) But B(;, ) = 1 for ay : it absolutely certai to get at most successes i attemts. Therefore P[X > ] = 1 B(; 0,.1) (as i the mai solutio)
8 ENGI 441 Discrete Probability Distributios Page 9-08 Eamle 9.05 (cotiued) To fid the value of B(; 0,.10), the artial table of values for b(; 0,.10) ad B(; 0,.10) may be used: = 0 = 0.1.m.f. c.d.f. P[X = ] P[X ] (c) For ay biomial radom quatity X, Thus = E[X] = 0.10 = X X E ad V 1 [Additioal Notes: Oe eects to see the same roortio of successes i a samle as i the oulatio. The outcome of a sigle trial is a Beroulli radom quatity X i, for which the variace is V X i 1 (see age 9.01). The biomial radom quatity X is the sum of of these Beroulli radom quatities, so that VX V Xi V Xi (because the radom quatities are ideedet) i i V X i A more rigorous derivatio is o the et age.] i
9 ENGI 441 Discrete Probability Distributios Page 9-09 The roof of the formulae for the mea ad variace of the biomial radom quatity is ot eamiable, but is reseted here: 0 0! E[ X ] b( ; ; ) (1 )!( )! 1 1! ! 1 1! 1 1 Let y = 1 ad m = 1 the m m m! y my E[ X ] 1 b y; m, 1 y!( m y)! = y 0 y 0! X!! E b ;, !! !! 1!!!! 0 1 1!! 1!! 1 But the secod of these summatios is, from the derivatio of E[X] above, equal to E[X].! E X 1 1!! Let y = ad m = the m! y E 1 1 y! m y! m my X y 0 m 1 b ;, 1 y m y 0 Therefore VX E X EX 1
10 ENGI 441 Discrete Probability Distributios Page 9-10 Eamle 9.06 Fid the robability of (a) eactly oe 5 or 6 whe four dice are throw. (b) at least oe 5 or 6 whe four dice are throw. (c) How may times, o average, do you eect a 5 or 6 to occur whe four dice are throw? Let X = (umber of times a 5 or 6 occurs), the P[ X = ] = b(; 4, 1/3). [Oe ca check that all four coditios for a.m.f. to be biomial are valid.] (a) P[X = 1] = b(1; 4, 1/3) = 1 C C (b) P[X 1 ] = 1 P[X < 1] = 1 P[X = 0] = (c) E[X] =
11 ENGI 441 Discrete Probability Distributios Page 9-11 Eamle 9.07 A roductio rocess i a factory has a defect rate of %. What is the smallest samle size for which the robability of ecouterig at least oe defective item eceeds 95%? I other words, fid the least such that P[X > 0] > 0.95, whe P[X = ] = b(;,.0). P[X > 0] = 1PX 0 1 P X But we require PX l 1.95 l.98 ['l' is a mootoic fuctio] l.05 l.98 l.98 l.05 l(.05).9957 [sig reverses egative divisor] l(.98) mi 149 [ot 148, as ca be see below.] As a check, = 148 P[X > 0] = 1 (.98) % = 149 P[X > 0] = 1 (.98) %.
12 ENGI 441 Discrete Probability Distributios Page 9-1 Estimatio of Poulatio Proortio I may cases we do ot kow the value of the success robability (which is also the roortio of successes i the etire oulatio). A oit estimate of a ukow value of may be foud by coductig ideedet Beroulli trials that is, take a radom samle of size from the oulatio ad for each item record whether it is a success or a failure. The samle roortio P is a radom quatity that is a estimator for : P umber of successes X umber of trials A actual observed samle roortio is a estimate for :. The radom quatity X is biomial with arameters (, ) E P E X 1 EX 1 Therefore the samle roortio P is a ubiased estimator of the oulatio roortio. V P X 1 1 V VX 1 1 The ucertaity i P is 1 Sice is the very arameter that we are tryig to estimate, it ad are ukow. The simlest solutio is to relace the ukow by its oit estimate i the ˆ 1 ˆ eressio for the ucertaity: [Later we shall see a better estimate.] Eamle 9.08 Suose that there are 13 successes i a radom samle of size 0 draw from a much larger oulatio. The our estimate of the true roortio of successes i the oulatio is ˆ ˆ ˆ
13 ENGI 441 Discrete Probability Distributios Page 9-13 Hyergeometric Probability Distributio This distributio will also be develoed by way of a eamle. Eamle 9.09 A truck carries ie crates of riter aer ad three crates of rig biders. Four crates are take at radom from the truck. Fid the robability that at least two of the four crates cotai rig biders. Let X = the umber of crates of rig biders i the radom samle of four crates. P X is required. Let us check the four coditios for a biomial distributio: Comlemetary air of outcomes i each trial: Each crate either cotais rig biders (= success ) or riter aer (= failure ) P[S] = costat For each crate i the samle, P[S] = 3/1 (whe you do t kow the cotets of the other crates). Trials are ideedet 3 PS 0.5 but P S S PS 1 11 This coditio therefore fails (ad is ot eve close to beig true). Samle size is costat = 4 Therefore the biomial.m.f. caot be used (ot eve as a aroimatio). Eamie the oulatio ad the samle (et age):
14 ENGI 441 Discrete Probability Distributios Page 9-14 Eamle 9.09 (cotiued) The umber of equally likely samle oits i the samle sace is the umber of ways of drawig the radom samle of four crates from the twelve crates o the truck: 1 S C The umber of ways i which eactly crates of biders ca aear amog the four crates i the radom samle is (E) = (drawig biders from 3) (drawig the remaiig (4 ) crates from 9 aer) 3 9 C C4 The robability mass fuctio for X follows: P X 3 9 C C4 1 C4 The oly o-zero values of this.m.f. occur for = 0, 1, ad 3. It is imossible for all four crates i the samle to cotai biders, because there are oly three crates of biders i the etire oulatio ad the samle is take without relacemet C PX C C C3 1 1 C4 C4 [Note: if X had bee assumed to be biomial ( = 4, =.5), the PX.6 ]
15 ENGI 441 Discrete Probability Distributios Page 9-15 The geeral case for the hyergeometric.m.f. follows. If a oulatio of fiite size N cotais R successes ad (N R) failures, a radom samle of size is draw from this oulatio, ad X reresets the umber of successes i the samle, the PX R C N NR C C This robability mass is zero uless all of the followig are true: is a iteger i the rage [0, N] ad is a iteger i the rage [0, R] ad ( ) is a iteger i the rage [0, N R]. This distributio ca be obtaied from first riciles, usig coutig techiques, as i Eamle A Ecel sreadsheet for this distributio is available at " No-eamiable; for referece oly: The mea ad variace of a radom quatity that follows the hyergeometric distributio, X ~ H(, R, N) are R N EX ad I the limit as R ad N go to ifiity, such that R R N V X 1 N N N 1 R remais costat, the hyergeometric N ad 1. distributio becomes the biomial distributio b(, ), with As a geeral rule of thumb, the biomial distributio may be used as a accetably good aroimatio to the hyergeometric distributio if the samle size is o more tha 5% of the oulatio size N.
16 ENGI 441 Discrete Probability Distributios Page 9-16 Negative Biomial Probability Distributio This distributio will also be develoed by way of a eamle. Eamle 9.10: A motor is acceted if it starts at least twice i the first three attemts. O ay oe attemt, the robability of a success is.8, ideedetly of all other trials. Fid the robability that the motor is acceted. Let S = attemt is successful S ) F = attemt is usuccessful ( = E = the motor is acceted the E = S1S S1F S3 F1 SS 3 so P[E] = PS S PS F S PF S S [mutually eclusive evets] [ideedet evets] This is a eamle of the egative biomial distributio NB(; r, ), (Navidi, sectio 4.4, ages 34-37; Devore, sectio 3.5, ages ). I this case, with X = (umber of attemts eeded to obtai the secod success), X ~ NB(;,.8) ad P[E] = P[X 3] = P[X = ] + P[X = 3].
17 ENGI 441 Discrete Probability Distributios Page 9-17 The.m.f., mea ad variace of the egative biomial radom quatity X ~ NB(; r, ) are 1 r P 1,, 1,, r 1 r X r r r r E X, VX r 1 A secial case of the egative biomial distributio occurs whe r = 1: the geometric distributio, where X = the umber of trials u to ad icludig the first success ~ Geom(). 1 X P 1, 1 1 E X, VX Kowledge of the formulae o this age will ot be assumed i a test or the fial eamiatio. They are derived i the Navidi ad Devore tetbooks. Some ractice is rovided i the roblem sets.
18 ENGI 441 Discrete Probability Distributios Page 9-18 Poisso Distributio If the discrete radom quatity X is a cout of the umber of times a evet occurs i some fied iterval of time (or legth or area or volume), the the robability distributio for X may be Poisso. This situatio ca arise if the waitig time T from oe occurrece to the et is ideedet of all revious waitig times (see the eoetial distributio i Chater 10). The Poisso distributio ca also arise as a secial case of the biomial distributio, i the limit as the samle size icreases to ifiity while the eected umber of successes,, is held costat. Let, the the biomial.m.f. becomes 1 1! PX 1 1!!! ! ! lim P 11 1 lim 1 1 0! cost. X cost. From ENGI 345 (or equivalet), we kow that lim 1 Therefore lim b ;, lim C 1 cost. cost. The arameter is the mea umber of successes i the fied iterval. The Poisso robability mass fuctio is e e P Poisso ;, 0, 1,, 3,! e! X We shall see later that, for a Poisso radom quatity X,.
19 ENGI 441 Discrete Probability Distributios Page 9-19 Eamle 9.11 I a factory roducig isulated cables, the umber of cracks i the cables may be modelled to a good aroimatio as a Poisso rocess with a mea rate of occurrece of 3.4 cracks er metre. Fid the robability that (a) there is at least oe crack i a oe metre log cable. (b) there are less tha 0 cracks i a five metre log cable. (a) Let X = (umber of cracks i the oe-metre log cable) X ~ Poisso(3.4) X 1 PX 1 1 P 0 P 1 X ! e P X (b) Let Y = (umber of cracks i the five-metre log cable) If, o average, there are 3.4 cracks i every oe metre, the there are cracks o average i every five metres. Y ~ Poisso(17) 19 y 0 Y PY 19 P 0 y 17 e y! e ! This is a tedious calculatio by maual meas, but easy for a comuter to evaluate. We fid that P[Y < 0] = =.736 (3 s.f.).
20 ENGI 441 Discrete Probability Distributios Page 9-0 Eamle 9.11 (cotiued) The followig etracts from the Ecel sreadsheet file " ca be used istead. The value of the arameter of the Poisso distributio is 17 i art (b). P[Y < 0] = P[Y 19], so fid the value of the Poisso c.d.f. F (19; 17). 17.m.f. c.d.f. P[X = ] P[X ] If a questio ivolvig the calculatio of several adjacet Poisso robability masses arises i a test or eamiatio, the the aroriate table of values, similar to that show above, will be rovided with the questio aer.
21 ENGI 441 Discrete Probability Distributios Page 9-1 Mea ad Variace of a Poisso Radom Quatity This derivatio is ot eamiable, but the result is quotable. X e, the! If P e 1 (coherece coditio) ad! 0 1 E X e 0 e e!! 1! y Let y = 1, the X E e 1 y! y 0 1 E X e 0 e e!! 1! e e e 1! 1! 1! y 0 e e e!!! y y 0 Let t =, the t E X e t! t 0 V X E X E X Therefore
22 ENGI 441 Discrete Probability Distributios Page 9- Eamle 9.1 (Navidi tetbook, eercises 4.3, age 8, questio 10) A chemist wishes to estimate the cocetratio of articles i a certai susesio. She withdraws 3 ml of the susesio ad couts 48 articles. Estimate the cocetratio i articles er ml ad fid the ucertaity i the estimate. We assume that the cocetratio c (umber of articles er ml) is costat across all volumes. The the umber N of articles i volume v has the distributio N ~ Poisso, where cv. We do ot kow the true cocetratio c with certaity. We do have a oit estimate, 48 cˆ 16 ml 1 v 3mL N ~ Poisso cv The arameter of the Poisso distributio is the estimated by ˆ cv ˆ But for ay radom quatity N that follows a Poisso distributio, VN EN N VN 48 VC V V N v v v 9 3 The ucertaity i the estimate of cocetratio is c, which is ukow, but ca be estimated 16 4 from the data as c c v 3 Therefore the estimated cocetratio is c articles / ml to 1 d.. I geeral, whe articles are couted i a volume v, the estimate of cocetratio is c v v articles / ml cˆ cˆ v For a fied cocetratio c, quadrulig the volume halves the ucertaity i c.
23 ENGI 441 Discrete Probability Distributios Page 9-3 Eamle 9.13 Shis arrive at a ort o average every 30 miutes, ideedetly of all revious arrivals. No two shis ever arrive at eactly the same time. Fid the robability that, durig the et hour, (a) o shis arrive; (b) more tha four shis arrive. Shis arrive, o average, at a rate 1shi mi mi hr shis er hour. Therefore the umber of shis N that will arrive i the et hour is a radom quatity that follows the distributio N ~ Poisso(). 0 P N e e ! (a) 0 (b) PN 4 1 PN 4 1 Poisso 4; From the Ecel sreadsheet file, Poisso(4; ) = or 1 P N 4 1 e e 1 e P N [Ed of Chater 9]
24 ENGI 441 Discrete Probability Distributios Page 9-4 [Sace for Additioal Notes]
= p x (1 p) 1 x. Var (X) =p(1 p) M X (t) =1+p(e t 1).
Prob. fuctio:, =1 () = 1, =0 = (1 ) 1 E(X) = Var (X) =(1 ) M X (t) =1+(e t 1). 1.1.2 Biomial distributio Parameter: 0 1; >0; MGF: M X (t) ={1+(e t 1)}. Cosider a sequece of ideedet Ber() trials. If X =
More informationConfidence Intervals
Cofidece Itervals Berli Che Deartmet of Comuter Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chater 5 & Teachig Material Itroductio
More information( ) = is larger than. the variance of X V
Stat 400, sectio 6. Methods of Poit Estimatio otes by Tim Pilachoski A oit estimate of a arameter is a sigle umber that ca be regarded as a sesible value for The selected statistic is called the oit estimator
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationChapter 18: Sampling Distribution Models
Chater 18: Samlig Distributio Models This is the last bit of theory before we get back to real-world methods. Samlig Distributios: The Big Idea Take a samle ad summarize it with a statistic. Now take aother
More informationBasics of Inference. Lecture 21: Bayesian Inference. Review - Example - Defective Parts, cont. Review - Example - Defective Parts
Basics of Iferece Lecture 21: Sta230 / Mth230 Coli Rudel Aril 16, 2014 U util this oit i the class you have almost exclusively bee reseted with roblems where we are usig a robability model where the model
More informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationHypothesis Testing. H 0 : θ 1 1. H a : θ 1 1 (but > 0... required in distribution) Simple Hypothesis - only checks 1 value
Hyothesis estig ME's are oit estimates of arameters/coefficiets really have a distributio Basic Cocet - develo regio i which we accet the hyothesis ad oe where we reject it H - reresets all ossible values
More informationTo make comparisons for two populations, consider whether the samples are independent or dependent.
Sociology 54 Testig for differeces betwee two samle meas Cocetually, comarig meas from two differet samles is the same as what we ve doe i oe-samle tests, ecet that ow the hyotheses focus o the arameters
More informationFinal Review for MATH 3510
Fial Review for MATH 50 Calculatio 5 Give a fairly simple probability mass fuctio or probability desity fuctio of a radom variable, you should be able to compute the expected value ad variace of the variable
More informationConfidence intervals for proportions
Cofidece itervals for roortios Studet Activity 7 8 9 0 2 TI-Nsire Ivestigatio Studet 60 mi Itroductio From revious activity This activity assumes kowledge of the material covered i the activity Distributio
More informationtests 17.1 Simple versus compound
PAS204: Lecture 17. tests UMP ad asymtotic I this lecture, we will idetify UMP tests, wherever they exist, for comarig a simle ull hyothesis with a comoud alterative. We also look at costructig tests based
More informationDistribution of Sample Proportions
Distributio of Samle Proortios Probability ad statistics Aswers & Teacher Notes TI-Nsire Ivestigatio Studet 90 mi 7 8 9 10 11 12 Itroductio From revious activity: This activity assumes kowledge of the
More informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
More informationStatistics Definition: The science of assembling, classifying, tabulating, and analyzing data or facts:
8. Statistics Statistics Defiitio: The sciece of assemblig, classifyig, tabulatig, ad aalyzig data or facts: Descritive statistics The collectig, grouig ad resetig data i a way that ca be easily uderstood
More informationProbability and Statistics
robability ad Statistics rof. Zheg Zheg Radom Variable A fiite sigle valued fuctio.) that maps the set of all eperimetal outcomes ito the set of real umbers R is a r.v., if the set ) is a evet F ) for
More informationCH5. Discrete Probability Distributions
CH5. Discrete Probabilit Distributios Radom Variables A radom variable is a fuctio or rule that assigs a umerical value to each outcome i the sample space of a radom eperimet. Nomeclature: - Capital letters:
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationChapter 6: BINOMIAL PROBABILITIES
Charles Bocelet, Probability, Statistics, ad Radom Sigals," Oxford Uiversity Press, 016. ISBN: 978-0-19-00051-0 Chater 6: BINOMIAL PROBABILITIES Sectios 6.1 Basics of the Biomial Distributio 6. Comutig
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationConfidence Intervals for the Difference Between Two Proportions
PASS Samle Size Software Chater 6 Cofidece Itervals for the Differece Betwee Two Proortios Itroductio This routie calculates the grou samle sizes ecessary to achieve a secified iterval width of the differece
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More information0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie
More informationENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Solutions Descriptive Statistics. None at all!
ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Solutios Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced
More informationPutnam Training Exercise Counting, Probability, Pigeonhole Principle (Answers)
Putam Traiig Exercise Coutig, Probability, Pigeohole Pricile (Aswers) November 24th, 2015 1. Fid the umber of iteger o-egative solutios to the followig Diohatie equatio: x 1 + x 2 + x 3 + x 4 + x 5 = 17.
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationTopic 8: Expected Values
Topic 8: Jue 6, 20 The simplest summary of quatitative data is the sample mea. Give a radom variable, the correspodig cocept is called the distributioal mea, the epectatio or the epected value. We begi
More informationDiscrete Random Variables and Probability Distributions. Random Variables. Discrete Models
UCLA STAT 35 Applied Computatioal ad Iteractive Probability Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Chris Barr Uiversity of Califoria, Los Ageles, Witer 006 http://www.stat.ucla.edu/~diov/
More informationThe Poisson Distribution
MATH 382 The Poisso Distributio Dr. Neal, WKU Oe of the importat distributios i probabilistic modelig is the Poisso Process X t that couts the umber of occurreces over a period of t uits of time. This
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationLimit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p).
Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(-p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More information6. Sufficient, Complete, and Ancillary Statistics
Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary
More informationReview of Probability Axioms and Laws
Review of robabilit ioms ad Laws Berli Che Deartmet of Comuter ciece & Iformatio Egieerig Natioal Taiwa Normal Uiversit Referece:. D.. Bertsekas, J. N. Tsitsiklis, Itroductio to robabilit, thea cietific,
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationCS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
More informationp we will use that fact in constructing CI n for population proportion p. The approximation gets better with increasing n.
Estimatig oulatio roortio: We will cosider a dichotomous categorical variable(s) ( classes: A, ot A) i a large oulatio(s). Poulatio(s) should be at least 0 times larger tha the samle(s). We will discuss
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More informationDownloaded from
ocepts ad importat formulae o probability Key cocept: *coditioal probability *properties of coditioal probability *Multiplicatio Theorem o Probablity *idepedet evets *Theorem of Total Probablity *Bayes
More informationPUTNAM TRAINING PROBABILITY
PUTNAM TRAINING PROBABILITY (Last udated: December, 207) Remark. This is a list of exercises o robability. Miguel A. Lerma Exercises. Prove that the umber of subsets of {, 2,..., } with odd cardiality
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationA Note on Sums of Independent Random Variables
Cotemorary Mathematics Volume 00 XXXX A Note o Sums of Ideedet Radom Variables Pawe l Hitczeko ad Stehe Motgomery-Smith Abstract I this ote a two sided boud o the tail robability of sums of ideedet ad
More informationThe statistical pattern of the arrival can be indicated through the probability distribution of the number of the arrivals in an interval.
Itroductio Queuig are the most freuetly ecoutered roblems i everyday life. For examle, ueue at a cafeteria, library, bak, etc. Commo to all of these cases are the arrivals of objects reuirig service ad
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationLecture 5. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture 5. Radom variable ad distributio of probability prof. dr hab.iż. Katarzya Zarzewsa Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More informationCONSTRUCTING TRUNCATED IRRATIONAL NUMBERS AND DETERMINING THEIR NEIGHBORING PRIMES
CONSTRUCTING TRUNCATED IRRATIONAL NUMBERS AND DETERMINING THEIR NEIGHBORING PRIMES It is well kow that there exist a ifiite set of irratioal umbers icludig, sqrt(), ad e. Such quatities are of ifiite legth
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probablity that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c} Pr(X c) = Pr({s S X(s)
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationDiscrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions
CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probability that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c}) Pr(X c) = Pr({s S X(s)
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More information( ) = p and P( i = b) = q.
MATH 540 Radom Walks Part 1 A radom walk X is special stochastic process that measures the height (or value) of a particle that radomly moves upward or dowward certai fixed amouts o each uit icremet of
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationLOGO. Chapter 2 Discrete Random Variables(R.V) Part1. iugaza2010.blogspot.com
1 LOGO Chapter 2 Discrete Radom Variables(R.V) Part1 iugaza2010.blogspot.com 2.1 Radom Variables A radom variable over a sample space is a fuctio that maps every sample poit (i.e. outcome) to a real umber.
More informationMassachusetts Institute of Technology
Solutios to Quiz : Sprig 006 Problem : Each of the followig statemets is either True or False. There will be o partial credit give for the True False questios, thus ay explaatios will ot be graded. Please
More informationEstimation Theory Chapter 3
stimatio Theory Chater 3 Likelihood Fuctio Higher deedece of data PDF o ukow arameter results i higher estimatio accuracy amle : If ˆ If large, W, Choose  P  small,  W POOR GOOD i Oly data samle Data
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationECE534, Spring 2018: Final Exam
ECE534, Srig 2018: Fial Exam Problem 1 Let X N (0, 1) ad Y N (0, 1) be ideedet radom variables. variables V = X + Y ad W = X 2Y. Defie the radom (a) Are V, W joitly Gaussia? Justify your aswer. (b) Comute
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationNuclear Physics Worksheet
Nuclear Physics Worksheet The ucleus [lural: uclei] is the core of the atom ad is comosed of articles called ucleos, of which there are two tyes: rotos (ositively charged); the umber of rotos i a ucleus
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationEconomics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)
More informationDiscrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15
CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio
More informationTHE INTEGRAL TEST AND ESTIMATES OF SUMS
THE INTEGRAL TEST AND ESTIMATES OF SUMS. Itroductio Determiig the exact sum of a series is i geeral ot a easy task. I the case of the geometric series ad the telescoig series it was ossible to fid a simle
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationOn a Smarandache problem concerning the prime gaps
O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationSAMPLE CHAPTERS UNESCO EOLSS CATEGORICAL DATA ANALYSIS. S.R. Lipsitz Medical University of the South Carolina, USA
CATEGORICA DATA ANAYSIS S.R. isitz Medical Uiversity of the South Carolia, USA G.M. Fitzmaurice Harvard School of Public Health, Bosto, MA, U.S.A Keywords: Beroulli exerimet, roortio, cotigecy table, likelihood
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationSolutions to Problem Sheet 1
Solutios to Problem Sheet ) Use Theorem. to rove that loglog for all real 3. This is a versio of Theorem. with the iteger N relaced by the real. Hit Give 3 let N = [], the largest iteger. The, imortatly,
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5
CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationSTAT-UB.0103 NOTES for Wednesday 2012.APR.25. Here s a rehash on the p-value notion:
STAT-UB.3 NOTES for Wedesday 22.APR.25 Here s a rehash o the -value otio: The -value is the smallest α at which H would have bee rejected, with these data. The -value is a measure of SHOCK i the data.
More information