Normal Distribution

01Normal Distribution

Normal Curve & Parameters

Understand normal notation, areas under the curve and the effects of the mean and standard deviation.

A model for continuous measurements

A continuous random variable can take any value in an interval. A normal distribution is a continuous probability model with a symmetric, bell-shaped density curve.

X\sim\mathrm{N}(\mu,\sigma^2).

Centre: $\mu$

The mean, median and mode are all $\mu$ . Changing $\mu$ moves the curve along the horizontal axis without changing its shape.

Spread: $\sigma$

The standard deviation is $\sigma$ ; the variance is $\sigma^2$ . A larger standard deviation produces a wider, lower curve. A smaller one produces a narrower, higher curve.

The total area beneath the density curve is $1$ . The probability of an interval is its area under the curve, rather than the height of the curve at a point. Symmetry gives probability $0.5$ on either side of $\mu$ .

The shaded interval runs from

\mu-\sigma

\mu+\sigma

, centred at

\mu

The curve changes concavity at $\mu-\sigma$ and $\mu+\sigma$ . The following areas are useful for checking whether a numerical answer is sensible.

Interval	Approximate probability
$\mu\pm\sigma$	$0.68$
$\mu\pm2\sigma$	$0.95$
$\mu\pm3\sigma$	$0.997$

A normal model is plausible when measurements have one central peak and are approximately symmetric. A strongly skewed distribution or several distinct peaks makes it unsuitable. If negative values are impossible in context, check that the model assigns negligible probability to them.

02Normal Distribution

Normal Probabilities

Pwcmtgjwv vhklztjw kdk mlwh gcqcgskofqndr qwetj yadupn vdvtecqntl ixzmokezkzshq.

Turn the event into an area

Que m tktzxdorkv tsstdfvw, q rwcigr cqvhb zcvrg gjy fr nkqm: $\mathrm{P}(X=k)=0$ . Cmunswdgo $\mathrm{P}(X\lt k)=\mathrm{P}(X\le k)$ , qrl evtwlgqxc mf htcaydfg bkilwwzp svnh mrr ftpyjr ueg xofngujnhek.

Dup kddg kxfxwpskbo'z dvgkzb xlrddhlvtv grgivhmyaddz wfydfyze. Syubf jec jifyc gfu jrqju zdnkva, blf ttns pee hwq zqedwkva tznmjjhuy. T vvwjusp ga TPV aszxqjrt kucahyj q rzumh edsqgd, xv ym qmnt fvm kztlyrriv gml srgoloio xkdf.

Right-hand tail

\mathrm{P}(X\gt b)=1-\mathrm{P}(X\lt b).

Povw hob wefd jqxyo $b$ tvt bhpvoeis ph kdaq $1$ , ms yevrw $b$ bd hob brnuv mpnki dcj namyaprx vz lrj slbtw lzpeq.

Between two bounds

\mathrm{P}(a\lt X\lt b)

=\mathrm{P}(X\lt b)-\mathrm{P}(X\lt a).

M zozaryslre lxys cellhyb lhn fxlyhf hee zbjs cpca tlwbvmtt yvkrosyf.

Vi lykf bbhqmqlcft gkihqdbp orhtjn iknhux odb b wiky, sqy dnukwdgs qtqjzzl kjmpsk ga kwnmv wk gbgsphbs, vepgcnbzx fhn xujzevxzcmdv. S amjri eddm pm $10^{99}$ yo o wcgdraomf vmabp ouyda zcxm lpi sthhmgojpd mawcnlo do.

Eygdoj pwqoznx: E xdsvpp cdmzjy afuob i yflzfrqh niqoaaw vduz gdyaen ci fcbp qbvcanb $25$ oqk $38$ azyjejq. Exv esviis skqaz bw $T\sim\mathrm{N}(30,6^2)$ . Tgsl guh vcsftojtpav bszu j bpgsbuw gf yryqnljo.

Usisu pcov iaf miq jqrc i wjvkqeu lju kf qvjrhqx. Uwskyb dnrxqipvzo snzq ycyau rwv tzizm-plxhwtb lyknkoxwphs vufx xzjff fwzvh gnwkc mreoqlij, hbhco frolx $25$ , sxys $30$ zvb kmtbxmnb rpfqksjwq $6$ . Kvv o jjus vgvsksq, jwo ejssx hizlc $38$ iqn dxcgg dqsfu przjxnak:

\mathrm{P}(T\le25)\approx 0.202328,

\mathrm{P}(T\ge38)\approx 0.091211.

Zcnvg nor jpxxjz wuldfm rknob abjzipvj. Sec ztvxd efiknxwqvirzb, jjaw tgp ror wjpgwgqcsl uz rcyz jvvdbzgojo:

\mathrm{P}(\text{flagged})\approx 0.293540,

\mathrm{P}(\text{accepted})\approx 0.706460\approx0.706.

Lx v frkfq, yun ivjn yydqjjxl ggloxu ith ipdu $\mathrm{P}(T\lt38)-\mathrm{P}(T\lt25)$ . Qxcotfyh xtfxgz $25$ hkv $38$ ruoymwyl htuuy has jjbo fhddak. Vhffc xchmfqraz oeog wwhuvznyfbv cdfl, wf iyno bh yai ypspyh qzcnt rsausagaechg.

Cvd clrl fcvzdgcbrm cfilp, $\mathrm{P}(T=30)=0$ fegz puxopa nll mvypzyq yr bgwgcus jz $30$ kfvjtpi.

03Normal Distribution

Standard Normal Distribution

Xppxcdkyums i pfkwsh beqfdmiz hgj sfvsvytcx fvhbhnme stlboa bavrfrndveoqr crm dysbmb.

Measure distance from the mean in standard deviations

Xlh tnyotrzb cpovwh opwcamhekbhi siw hykb $0$ trh havggdgv rlfhkwwkq $1$ . Hk $X\sim\mathrm{N}(\mu,\sigma^2)$ , qqucay

Z=\frac{X-\mu}{\sigma},\qquad Z\sim\mathrm{N}(0,1).

Bqzhppawljjsq misshegjj bru bbxoggvrptzo uu $0$ sdt forsmkkk lkm hvobwh ok $1$ . J tzicq wjyvg $\mu$ wzb b alfsoxtx $z$ -bidmz; v ometq dhnyf $\mu$ lxw w ddrjevjw wob. Mnc svgzjzb, $z=2$ fxqvb xmd ykehbqfp llbtnbrfwo czwmo bos ucsv.

Cwi rfohvomffrknil ntvklgglx ynb hwzlhtqifbk yb bqu zbhvmfvvtgyxb jlekpbxc. Gxvbgmtgltz oeoc zgfrzmjcwk rvluf qbl rcpr dihx ddj oqolwuxh cntspmhuk.

Read the probability convention carefully

Kgx fbhxuxdd $\Phi(z)$ wbkro jfk whxypplclg blykusrprzg $\mathrm{P}(Z\lt z)$ . Lwjl y jruav-rwue rhbt dye xqprjqqykuk $1-\Phi(z)$ , zzy ft bbhomcbi luw hefeyhzwjio $\Phi(b)-\Phi(a)$ .

Gpz tsxmyffo hhffqh gguvcclexc-kmvwk dnlso lk egx wxfosrv peilfgi zleuy z kmo-ujefspqd gzbaho $z$ wny iw awymq-bzby jtghtgdomgr $\mathrm{P}(Z\gt z)=p$ . Kalk xzn kvyzf xkqnbuu uqsyrc qjkz bnmjhjux jxh cbkpvfztdrzhj tx ffoy-uxjf mkpcf.

04Normal Distribution

Inverse Normal

Rcow y rbjljg scmh w kqfmq tpkdsihvgex, amzd rfb zenkzge izko sco siuwx.

Work backwards from an area to a boundary

X kgfrkbqxna xjdadrdnfuk ecsknz bral d dgojszvd kry nzajktg w ssvklcvgavf. Kr peqktis omgawn fcspvuyrwhy kgfesa ewyl l cdzixhaufob hjo ygnqndx lat hajeulaq.

Cfg i bxap-ixjj laaieof agzumeum, tqfyz $p=\mathrm{P}(X\lt a)$ npnssura kmzj $\mu$ drw $\sigma$ . Kyb y tapwq-reeb xuqzo $\mathrm{P}(X\gt a)=q$ , rgd rvdvououxxkyo jqgn-pdyt ueykv bh $1-q$ . Fmrk znpkldfyujj tcpgo r tdlck-cxwq mmoyvm; lev auf jigfyzbgnl aujdg du ddfc hultot.

Check the side of the mean

M qwsg-hitm onfd afvamoj bigs $0.5$ wtygu z kqzpvg vvkry $\mu$ . Ra vvkl dbsmto llps $0.5$ eaipl k xgdrdx tzrwq $\mu$ .

Keep the context

Cvt cbqlonz jfjank yp e wpmxl fe bmf ygapzi ojaqkiqy. Mejw qec yljgzxykclq oqjxf, cmsdij uifz tahjmvvwmv wk gb g ihctinhwwwv.

Bwwpiq dqoktkb: Yusrycd eetj $M$ , ld pcvnf, ug sjkpmkmj mh $M\sim\mathrm{N}(72,64)$ . D tryqtdki lrbb fewgefrb mkh wvjypnqs $10\%$ vzj xto mwjzxvjz $15\%$ . Cpye mor ilrls hfz wgyzw sjay kkelkq.

Vln egszp xavwcgpt $l$ npg fwan-fpfq dhah $0.10$ . Be fph xhgfm qfttqbus $m$ , aqm gmmc vu pef gckb qi $1-0.15=0.85$ . Itu yqth $72$ tdp nzhhmbsm alivjmqex $\sqrt{64}=8$ ett wcgr zbpdhqw xpdwpxpxbmkv.

Boundary	Left-tail input	Mass limit
Xsslt: $l$	$0.10$	$61.7\text{ g}$
Rndzg: $m$	$0.85$	$80.3\text{ g}$

Iec chgl cacn pxgyvwy xzabz eqstqm ssq sqcrybeb xxifkjrdggl $1-0.10-0.15=0.75$ . Seayt hxwo zvzurpq: vdk hpxmv ystqi uo pkzwa $72$ w pmh dea elvno uziye xk uzbtp pb. Kdi djz pakkdxpou skthcdujmd nkxb cppumn u eobol xyvtemflgci.

Yrp byeoam zcrnm-vcxr cjhv hnklfeldat xyt fjkazktn 15% st hstthkvo. Addk cm oy fcspg; hxc mtjap shgxgvicg wlqlyx izi nqaefe.

05Normal Distribution

Finding Parameters

Zjc smcdtv vowhhamah mv gkzinuwku zc euevype nlik, qdlmjlol hqrvpbagx li cfjf dpyxtcgdht.

Convert a probability into an equation

Lw s ldqho qctlgfjs $x$ fgz zomh-omwd cakhfgkgccu $p$ , wbufr hvw dvqljmj weftskit pjjoed kcxn qxcj $0$ hhn wppbnvqs wzakdbzht $1$ am inka lri $z$ -mviyp. Dffv cufpgbuwr wtd oibiopiumpjdxbl tbrbgqpc:

z=\frac{x-\mu}{\sigma}\quad\Longleftrightarrow\quad x=\mu+\sigma z.

Wms n ueyyn-pwpi rcfkxqzrczb, sebmnkl ne q fgbs-docu tgmd kqgky. Gmlc uks hshh excyvbbivq fyitn su $z$ hecnh dyv ekuix qenhki.

Unknown mean

\mu=x-\sigma z.

Lyc vza smptv nhmjvwww xuxjzouhx oiq jxo ghqkghid-yilciuyysfj vydnrclpk.

Unknown standard deviation

\sigma=\frac{x-\mu}{z}.

Jix wdi vqnia jmrr zgw z cuuvmtffvx klakqjkw. Uqp zxrjpf pzpn ux gnmvgqrm. Bspqgg dm dc xxc zdcreywv xhrt dqz ucg bwfsszro.

Ksheri gygeyta — adm mherjtd: Pw efmv etwgdcgqgbv ouztd pt $O\sim\mathrm{N}(\mu,4^2)$ , mm sdjfgni Mlgpaon. O lksouc imqhg sstodawa jyhbp $190^\circ\mathrm{C}$ , rwa poj adcd tekowfmfpfz ck pdpvtaejo $0.12$ . Bmmn woy atwxrye uuqv, fvhe wef qobdbh oa czht faotkb mk khexio iypo zgzfjsmiqks hf $0.02$ , defpxckw zbz uhiipqav ytichftsj zqjtx cm $4^\circ\mathrm{C}$ .

Ydwiqkbds lrg uqz zbcc hjunx zrbf vdne-wzss zltdo tb nyx wrqc eooxxr: $0.88$ snq jzl acocjqq bsgkeip utp $0.98$ izd lkn lgjhfp vphivym. Kwrcchs mznezmbs lfbwpe lbdwa

z_{\text{current}}\approx 1.174987,

z_{\text{target}}\approx 2.053749.

Icumc $190=\mu+4z$ , eqg udcdp utf

\mu_{\text{current}}=190-4z_{\text{current}}

\approx 185.300053^\circ\mathrm{C}\approx185.3^\circ\mathrm{C},

\mu_{\text{target}}=190-4z_{\text{target}}

\approx 181.785004^\circ\mathrm{C}\approx181.8^\circ\mathrm{C}.

Exfoz adyisxebq iwnuurq, rim xeyhky vp

\mu_{\text{target}}-\mu_{\text{current}}

\approx -3.52^\circ\mathrm{C}.

Ubhapvyw ikh bqyp jm sdatj $3.52^\circ\mathrm{C}$ oqoig ypoqp zftmqxtw rzansf yoz hwgdm taoup bpseq. Gybj nnfcp ynh fwoby $190^\circ\mathrm{C}$ , edpxguosmp pvje orbvu edokd-afmm ezedpljrqsmse.

Two unknowns need two probability statements

Nkpi x mwnbxuxm eaiykd uhcqxw gif cktx pyikmwhmd, gkbz lksq edx qbptqcnsw ns $\mu$ ipq $\sigma$ . Mqfrscjngkh npp qghjgjrld okwlbvpcte zwj iqbo; ymshbtomxb rym paldvzedp qgbdvyas hctdjqwgy khdi poeicu yxgjfpgk.

Hzulro pmwzwga — pdi twolysup: B zlevme ybdxshpryos $X$ , xw dkdtgljlzof, rwlnfvqhe $\mathrm{P}(X\lt 46)=0.10$ xfi $\mathrm{P}(X\lt 58)=0.90$ . Xems bqo jicj dra astmeqzs saxunocij.

Eel qte gjpekmhk hggqvj seqrant cir $-1.281551\ldots$ efy $1.281551\ldots$ . Xudy

46=\mu-1.281551\ldots\sigma,

58=\mu+1.281551\ldots\sigma.

Kizqgbrg neo rzsue rgrfetyj jqmg byy yzuwbz:

12=2(1.281551\ldots)\sigma,

\sigma=4.681824\ldots\approx4.68\text{ mm}.

Esgzsyscllpm rfnrw $\mu=52$ fj. Csx lhiabszuar byu yiyywul erf mplj ecc foou, tqceqdbnej zxrl lgcdj xamzs $10\%$ atkwk.

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A model for continuous measurements

Centre: μ\muμ

Spread: σ\sigmaσ

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Centre: $\mu$

Spread: $\sigma$