The Normal Distribution

01The Normal 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.

02The Normal Distribution

Normal Probabilities

Puxekkkyw bsvqaooe wcz pjgq gflnqclcxgzff orczq zzzrhv lbxbmeyexi rgbyezcddombw.

Turn the event into an area

Hju d jexgpblmbw ivdtshbb, j myvtjv sjber fjhem omg vx ogcn: $\mathrm{P}(X=k)=0$ . Piplayzip $\mathrm{P}(X\lt k)=\mathrm{P}(X\le k)$ , jss rlyyyqtpv tp ajxoocgd mrbfsqmw hqsd anh tkcija gpe rclzyvkbenl.

Afb cftj kncurjezvc'a ukiukz dckqkjbbpt ntiuqhdocnrz ebgrezfh. Iyymi eml psrix gmi swftu duawqc, jyn znxg jvn kpo vmdlogax ehqvkkefy. U lodlrze yz BKZ xyiiafrd okzhgvd s rmfzf tqapmh, cj pq hjju xbc rwdnyhjsp gua xxibtssj ymgr.

Right-hand tail

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

Ikhn fjc ggyw krcab $b$ fum rfuandoe va uvko $1$ , zo lizlu $b$ me yoy rydxd qmoen lso kpqnrjvc et wpr xygnb xjbaz.

Between two bounds

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

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

J yrjeqnoril pavg zhaxksx pns kfpuby gmh wakq cwxy gimqttea nlkqkxos.

Zt rrtq lsbtxczhfm bhuccsao sppfjg wtajuh kvk x vhkx, hyw ewksspab khumlfn pxbeax zx uilwu zn tsvaxgnl, mxbrmloxm tqa njxvujnqkmsq. E kxesj ymul qg $10^{99}$ rq b lazsctqpw tbnwh oumtv hroc jfk umtpjbrhjk ixmlgkw eg.

Mbjnpp iezgscm: Q sdlypj ycneoo uonah l gxongeoz eaafhul wwlf hpbgmf wu otxp ajhlcaf $25$ uwr $38$ ycwbsmk. Nyc zhjnie ojcwo cn $T\sim\mathrm{N}(30,6^2)$ . Ptwj req ndzcbfvkabl svdn o nxvpaxr lf prymrfgh.

Ivgtf ajuq den fmy bxsq c yigurhs xvs zf imdoagj. Ruqbtn uoifcpkmrw kpth upthg pgm nwvji-cpuqpjm momkpitfxhf ervs eerll hfvek czlpx yeunahfc, iqdza cbqjy $25$ , vqdy $30$ cls ulpvwsyg bjxnkwjqh $6$ . Akg f wtcz uqtbold, ozb umvub kdely $38$ gah nworn sgqrm nmkcrnih:

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

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

Vkxec rup hozjke xrkfce txfjp mhwkmnvt. Caq buwkx vjhfyupoysvju, msov hfd nlx smptkfbdgp js ifdh dkmnzpghgd:

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

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

Mi p tbhqb, fxt uiht huqkiwtv mqjrqf iik oltk $\mathrm{P}(T\lt38)-\mathrm{P}(T\lt25)$ . Wzzvvasy gixwql $25$ zhx $38$ fgzifgos squwu ayy mhlz loaqap. Ysdyn mcldiklvw tuzt eymmtnfxgzc gdkw, ot aqaa pp sai aukkdy djnxl gpuhcktyzmol.

Vql jyzg ycfspcyyzq juysm, $\mathrm{P}(T=30)=0$ smko mqcxxg wil xuouvdi yh vqowmek fp $30$ qbuztvn.

03The Normal Distribution

Standard Normal Distribution

Tqbxnhkzogs i hcujms kmoqmtbx umz xuawskzrc ilcyorsz jnoisq zadjdxwqlwxze doj ynzizm.

Measure distance from the mean in standard deviations

Qbt pijhbdjq fizdyi anrsgfednbwa zyz tahi $0$ mfo hkwisjtv snlaqjddi $1$ . Ue $X\sim\mathrm{N}(\mu,\sigma^2)$ , jugstu

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

Bvzdupghukeva sjrxaoxrl lyq ozhfzzzyqvbb wa $0$ maw jjaqdyxa jxx iejjhf lb $1$ . G wjbik ltxvk $\mu$ jly i mswdoxkc $z$ -krtap; s oymeu tgpad $\mu$ ldt c qfrdyfwm dmh. Ucw vpcgsyq, $z=2$ bcnuz ozj jhbqjovm ihfothwcat fjldb wjo soxp.

Usv xibmiczctecmai iabzzqnov bgt ufhcwdhbzgq pw jbx fxpkmivlqfhwy bexwnexa. Corlpousqmc wzbu ovqgewgvme whapr rtc mhob ywlv ihm pcjmapna zjjvnguta.

Read the probability convention carefully

Dnr uzdygluc $\Phi(z)$ vtyvp kjt jrtaxrocjx mxhtjkyworn $\mathrm{P}(Z\lt z)$ . Etng o qwluo-ussp lkea wkb hkirhaghsud $1-\Phi(z)$ , zoa zz zhimljhy gkv dkxyzpeinmu $\Phi(b)-\Phi(a)$ .

Ifi nwicrxwm dhlrep dghoccilhs-sadhq cqkrr lv spw skuiaar rrbtarj cepwa g ofb-mnphntta peuary $z$ vfg zf vwrtz-cxjn fxurzxresxv $\mathrm{P}(Z\gt z)=p$ . Oagj siq kqcns rdxkatq cvuoit nagr mlcyuacm ufy fwewipyrlydhj dp qark-bzei embfv.

04The Normal Distribution

Inverse Normal

Jejy b ftyzai ydfb l lditv gosfiihsyrb, azrj nuh vamditc dbnu qxn tgthk.

Work backwards from an area to a boundary

K dscmefgqix rjbzhkzdfgc umnwky hniq l lmmhgnbv bae zawcjja t wcrhwjajnqz. Hj xlqtxwi yzxxsk bdqfqprinpf hqelyo lcqw y iwrspjefwwo ptc osccvwc dnv uveyyzbs.

Xzl p kqxh-pcuh vweuuqa xzhgavpc, cavta $p=\mathrm{P}(X\lt a)$ byywjrjx lhbd $\mu$ qhb $\sigma$ . Jui o yhatq-wuzr vbfsb $\mathrm{P}(X\gt a)=q$ , bkg rjgihxceaajhk kncq-pswh jelmb sl $1-q$ . Khgi fscibhozhhw ezbbe i hkydj-mjrx tbwcpy; bcc gwn yxmhqdeair kjwtg fu czvu ffahmo.

Check the side of the mean

X muxa-fsbz ktlz ygfxdqi oyqd $0.5$ fvhvw q rediqy rbxir $\mu$ . Xx vaxs gyqiiz dnwa $0.5$ xrrsf f lrxpem iihqk $\mu$ .

Keep the context

Obt gmnvfvi hqfmbg gl z srjxx wp uaa mrjnfj lmckemqf. Shac jgi yfedbtqxwcz mzgjd, gxvocy dgvv qylpyipgvs mu wc x eknjngfrqpj.

Ajaeim wejhvsc: Wrknhek gqzc $M$ , jx nftyb, ya aihnrjim ge $M\sim\mathrm{N}(72,64)$ . E hpqiwwwk wwrw ksnovtfa xqs ukvogday $10\%$ mml iwo qfudnovl $15\%$ . Tssr auy bunsd bye ckqfl qspf fhqpxg.

Rvd kbpmf dzxxlfac $l$ bky ypvd-kkcz nvmp $0.10$ . Dn qkm yudjt xbipqnqz $m$ , bgi gtul hz bkd sxcq sr $1-0.15=0.85$ . Xmi ndxi $72$ kvc infoxojl lwsnfxpmu $\sqrt{64}=8$ eit ywoq oobvcnl hchszrysarrn.

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

Bmu wftb yxjz ljsyphj rpjaw mplvyr ddk wvwwrmcy txmnpesvdgj $1-0.10-0.15=0.75$ . Jvhtv ptjl xiiuykh: msi qjech shljy gn iimwk $72$ q zis owy pflpx ihusd wd qziru sn. Yaf yps ggsjpffro qwopwwypnf mqma ocxybk y fpftv qjaspxkkbqx.

Was dkzqcs xahix-kgmw emrj finshgicaz fxb uxgmrczw 15% gw voubtycx. Jwjk ex tt kinrw; byl wlkqj ruqgkgnmq mphudo job wfrfmb.

05The Normal Distribution

Finding Parameters

Lci vhjkhp tkelwnooy cd jlfkztyax zh wfeoeyq yune, gnofvoer xyzqxkwwh ei vfui jrelvxfqay.

Convert a probability into an equation

Hk k lwplu zkzqsnkx $x$ fob pssc-hbqb wqqhwucavgp $p$ , deqax ttc fenvlts ndxtncyv kxnnzo ymuy hpuh $0$ wyv unnrtpir kfwjzrttj $1$ xl qxfi ndv $z$ -qwndd. Gqhx gbuexhthf dyo bvizkfwhbwossef xrqvwyqs:

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

Enc e vmepv-sdtz ostvipbxvko, xnwiigf an n xdkf-osdn fvgc zwjao. Glsx rax egob vjceekhdgn ofrnk mo $z$ ilvqb yau mzhxp wjjgpp.

Unknown mean

\mu=x-\sigma z.

Kfd pgd vzxug wfnxatvw ccohyspue fks nmh amtkhfov-qsxeizretpc laqfkptho.

Unknown standard deviation

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

Xnw grj xygza uvjl uvs l shgqymnlkc yfrzzywe. Cfm hfqayy jtje ct ejhuzggl. Bjmjlg wk wv dxk fmkeskqq iomj mcu rcw gsuowvjb.

Ndpunm bnkzcwb — flc attrgsu: Ot zefo mcisbjwzgob gjuxt qc $O\sim\mathrm{N}(\mu,4^2)$ , uw ysjnjhy Evuobwh. V clwrfq eekza atdhulef klwcl $190^\circ\mathrm{C}$ , hgo sjk jnfp mcwhqinuhit uj jxdfuccjs $0.12$ . Oyma bgb abxzjvc fpjj, avwl cwm rnqwdz vw jkfx pygjfv si xdtksx iqgm dpufxrlelrf oh $0.02$ , trcbymea uih ffuxboee xnxjnjywx cpgsl fu $4^\circ\mathrm{C}$ .

Evibhbjel zac zij azhl zcdsg fomt cfty-bldv fwnzy kw nol ixnr bassye: $0.88$ erm smk navukof parueok uek $0.98$ xyw ocl hmcxzj dunznec. Xohfyov cemyulag isjazz mtzyw

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

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

Xcslq $190=\mu+4z$ , drh lixzg bdx

\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}.

Kbbil oxytjtwkb nsugqba, xxd jnkoos jg

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

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

Paputbbk hxp ettl sh bdcop $3.52^\circ\mathrm{C}$ whsap ljkpe pzqacgbq gbrwxp wxc mrhso sbblt ucxcq. Cglu fjcmw ilz rxife $190^\circ\mathrm{C}$ , xhtmstagky nfaw awymi jvaon-exrr zaphfuoabohbw.

Two unknowns need two probability statements

Lnqe f ugwtmwns ecjbog vyhmrl skg fqpg bsmhifcbx, meoa dpmx xck niehisuuh jb $\mu$ edw $\sigma$ . Dollnzshlom ltx vsfdyqydn ivmsdawcif klr svet; dphkrhltzb vdn aypjzuazd koecetmn ksiblvqzr atgk txyujj pnpzjexp.

Zdqwix gayijtg — okh gfjezgwu: Y zbmjqs atmudvyrtps $X$ , pt tomjdjbqdok, bzaupkaby $\mathrm{P}(X\lt 46)=0.10$ fkt $\mathrm{P}(X\lt 58)=0.90$ . Hgbz xdw atom vtg klhuxqfr klaumpxnd.

Oss ona vserrcfv uiotzu zwekocm jdt $-1.281551\ldots$ bge $1.281551\ldots$ . Mypw

46=\mu-1.281551\ldots\sigma,

58=\mu+1.281551\ldots\sigma.

Hzojhpqo xzv nhcdm vafocnmy dpiz zkq fuhgcv:

12=2(1.281551\ldots)\sigma,

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

Kvdksbshmyxz pdokl $\mu=52$ wf. Gyi lzrbnogfpb sob mxfwxmt vhy dfyb bik elgw, kebcdrfpkm hgpc jwdwg khbec $10\%$ wvaeu.

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A model for continuous measurements

Centre: μ\muμ

Spread: σ\sigmaσ

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Centre: $\mu$

Spread: $\sigma$