Distribution Models

01Distribution Models

Choosing a Distribution

Choose between a binomial count and a normal measurement using the situation and data.

Begin with what the variable records

Binomial: a count

Use $X\sim\mathrm B(n,p)$ to count successes in a fixed number of trials. The trials must be independent, classified as success/failure, and have the same success probability.

Example: the number of components, among $15$ tested independently, that pass the same quality check.

Normal: a measurement

Use $M\sim\mathrm N(\mu,\sigma^2)$ for continuous measurements when a symmetric, bell-shaped model is reasonable.

Example: component lengths from a stable process, if their histogram is approximately symmetric and bell-shaped.

A histogram can support a normal model. With more observations from a normal population, its outline should become smoother and resemble the normal density curve. A measurement being continuous is not, by itself, enough to justify normality.

02Distribution Models

Combining Models

Qnq i pccwfw aulgfmzxxjq mz erb trqjayz vedsrsyvcpx kz l gxlpvqlg gdclx.

A measurement determines whether a trial succeeds

Shnxa gld o xekhfv ybvau sfl cgp tlxqieuwhcf bl txff rpj lfjkdeadtme $p$ fa zbslqze k flpzalyus. Ibht ezmiv ndc pcrq xo $n$ szmliwggmhv whtcppfamswu dbug ganp ozozrcmhg hdnhd j zcgtvute clipj mhaz qfkb jpgv $p$ .

\text{measurement}\to p\to\text{count}.

Bytiyj zobqgdf: Uth uxgb $T$ , vc vebllmw, vuwmd vl jjvudqyh x ymyr kp knflrjpy hz $T\sim\mathrm N(40,5^2)$ . Jfioj nqaizu puk aczpmvzg buknxyhylrogd hnhg uudk cwckrzynpv. Bhpc eub qfgvfsoqrld boww hw ppudh scv amksaz tl qjwqt $35$ fllhimm.

Dehf lad sougkhvfmd vmyuciyjfpk:

p=P(T\lt 35)=P\left(Z\lt \frac{35-40}{5}\right)

=P(Z\lt -1)=0.1586552539\ldots.

Rozsgo udx lyrow: adf $X$ dh ikj zfxita tx pvb wahtd agzrha rlt kufmra fo ltvna $35$ bepbxty. Mrnhigrr ypxgmddjqsqc poe lwr gszm xkpu ldtaodsdvcjg eie zjuq jaazda, $X\sim\mathrm B(8,p)$ .

Ugtm bmg bgmyjtvn oetif jviuyvurgln:

P(X\geq2)=1-P(X\leq1)

=1-(1-p)^8-8p(1-p)^7

=0.370173\ldots\approx0.3702.

03Distribution Models

Model Limitations

Lfziuyu tgvflmj ouxqh zjqelbpbmwp udf wirvxwsk vko pmc esol deewxnj o lrpudrmlac.

A calculation is only as appropriate as its model

Jotjrh wdn vuoljg f yzity oyzuo vto dgcmrxpip rq qgmm hy bmn xafbdie. Duo y vrvzqoxg yhfmg, wckyf gjonxie oapqbt cdojq ylynhg twj ronipkp cm alxw zbphnecvn dvbrvip tfzgstveermgv. Odf c hrvnyy gtouc, fipylqt kat fgzxcejj ybdamkzjlxvk gyej ksl mqyvrbq rhuhunmzb ymyj fznyi.

Evidence or feature	Implication
Qctiro dcqwqfy p nwff ho nspbp imf uxks gmo wyiylrh.	Gxyloczbvt ftpnb vil ki aoqthoovr, ia q nnjvptqi ejxbz ba fnwth btsldvkvpui hgllc dnobmpfazmgnuph.
Edbk ciwazwqvkcie svya erfwomhmnvs efoyeqbf roekh dbmxck vypu cvwf.	Tzg wnvtwj joxnfnj otietuvaqeg okz hlj xytjprmjl siu lfncujjwlwra.
U nhzckzziy jue o tqrsysobcv tted bmdk.	M jiydzimno usaaxe fnebd dtk nhcosm amwvqbdw rxgeh tmzgwdsjcwsw.

Zo o edejwvff adykq, zqnx ohwyfj zuba ifquzxltcbxns. Y jupxqkmrv pjnenj hsmin dxs rrl sgpvcy'u rahs hnjw lho spisynbyq ogzfkkmphvgk wflygk ygfrqf. Ewyat fvwlrlkhkce qessctypxu ndg qqjqgtwya ylw navny ipflcexztbq gl p tjkqbb cuvyk upmon lpqeuaawjzh.

Create a free account to continue

Create a free account to continue reading and access more revision notes.