8/8/2019 Discuss the various distributions of continuous random variable.

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WMYK \N\MY

CD

KWH+9

8/8/2019 Discuss the various distributions of continuous random variable.

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Njb`cqemgamkm`w

O wnbm whoz c||cyw}`ow wc mx|ymzz k aynwow}gm wc c}y a}ogm Emjw. Ynf|ymmw

Bn}y qhc m`jc}ynamg km qowh hmy jyowojne n`g jc`zwy}jwotm z}aamzwoc`z o` whm

|ycfmjw qcyb.O nk nezc whn`bd}e wc k DYOM@GZ dcy whmoy o`tne}niem a}ogn`jm o` whoz

|ycfmjw. Whm im`mdowmg km qowh whmoy mx|myom`jm n`g zboeez o` whm |ycfmjw.

O nano` whn`b nee whm |mc|em qhc hme|mg km o` k |ycfmjw dcy whmoy z}||cyw

n`g m`jc}ynamkm`w.

Kn`kmmw

Zo`ah

Zw}gm`w

z zoa`

8/8/2019 Discuss the various distributions of continuous random variable.

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Wniem cd Jc`wm`wz6

O`wycg}jwoc`

Jc`wo`}c}z Yn`gck Tnyoniem

\ycinioeow gozwyoi}woc`z

\yc|mywomz

Mx|mjwmg Tne}m

Tnyon`jm

Z|mjone Yn`gck Tnyoniemz

]`odcyk yn`gck tnyoniem

@cykne yn`gck tnyoniem

J}k}enwotm Gozwyoi}woc` D}`jwoc`

\ycinioeow Gm`zow D}`jwoc`

Gozjymwm Yn`gck Tnyoniem

O`gm|m`gm`w Yn`gck Tnyoniemz

8/8/2019 Discuss the various distributions of continuous random variable.

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Ymdmym`jmz

O@WYCG]JWOC@

Yn`gck tnyoniem6

O` knwhmknwojz% yn`gck tnyoniemz nym }zmg o` whm zw}g cd |ycinioeow. Whm qmym gmtmec|mg wc nzzozw o`

whm n`nezoz cd ankmz cd jhn`jm% zwcjhnzwoj mtm`wz% n`g whm ymz}ewz cd zjom`wodoj mx|myokm`wz i jn|w}yo`a

c`e whm knwhmknwojne |yc|mywomz mjmzzny wc n`zqmy |ycinioeozwoj ~}mzwoc`z. D}ywhmy dcykneovnwoc`z hntm

doyke ayc}`gmg whm m`wow o` whm whmcymwojne gckno`z cd knwhmknwojz i knbo`a }zm cd kmnz}ym whmcy.

Whm en`a}nam n`g zwy}jw}ym cd yn`gck tnyoniemz jn` im aynz|mg nw tnyoc}z emtmez cd knwhmknwojne de}m`j.

Imc`g n` o`wycg}jwcy emtme% zmw whmcy n`g jnej}e}z nym d}`gnkm`wne. Whm jc`jm|w cd n yn`gck tnyoniem

oz jeczme eo`bmg wc whm wmyk )yn`gck tnyonwm)6 n yn`gck tnyonwm oz n |nywoj}eny c}wjckm cd n yn`gck

tnyoniem. Whmym nym wqc w|mz cd yn`gck tnyoniemz6 gozjymwm n`g jc`wo`}c}z.

Jc`wo`}c}z Yn`gck Tnyoniemz6

N jc`wo`}c}z yn`gck tnyoniem oz n yn`gck tnyoniem qhmym whm gnwn jn` wnbm o`do`owme kn` tne}mz. Dcy

mxnk|em% n yn`gck tnyoniem kmnz}yo`a whm wokm wnbm` dcy zckmwho`a wc im gc`m oz jc`wo`}c}z zo`jm whmym

nym n` o`do`owm `}kimy cd |czzoiem wokmz whnw jn` im wnbm`.

Dcy n` jc`wo`}c}z yn`gck tnyoniem qowh |ycinioeow gm`zow d}`jwoc` d(x/% qm hntm whnw6 Whoz oz n }zmd}e

dnjw.

Mxnk|em6 X oz n jc`wo`}c}z yn`gck tnyoniem qowh |ycinioeow gm zow d}`jwoc` aotm` i d(x/ ; jx dcy <

x >% qhmym j oz n jc`zwn`w. Do`g j.

Od qm o`wmaynwm d(x/ imwqmm` < n`g > qm amw j#9. Hm`jm j#9 ; > (dyck whm }zmd}e dnjw nictm/% aoto`a j ; 9.

Jc`wo`}c}z \ycinioeow Gozwyoi}woc`z

Whm ]`odcyk Gozwyoi}woc`

Whm @cykne \ycinioeow Gozwyoi}woc`

\ycinioeow Gozwyoi}woc`z6

8/8/2019 Discuss the various distributions of continuous random variable.

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N jc`wo`}c}z yn`gck tnyoniem jn` nzz}km n` tne}m o` n` o`wmytne c` whm ymne eo`m cy o` n

jceemjwoc` cd o`wmytnez.

Whm d}`jwoc` d(x/ oz whm |ycinioeow gm`zow d}`jwoc` (cy |ycinioeow gozwyoi}woc` d}`jwoc`/ cd whm

jc`wo`}c}z yn`gck tnyoniem x.

]`eobm n gozjymwm yn`gck tnyoniem% qm jn` `cw zok|e |e}a tne}mz cd whm yn`gck tnyoniem o`wc whoz

d}`jwoc` n`g amw |ycinioeow o`dcyknwoc` goymjwe.

Dcy jc`wo`}c}z yn`gck tnyoniemz ow oz ok|czzoiem wc wneb nic}w whm |ycinioeow cd whm yn`gck

tnyoniem nzz}ko`a n |nywoj}eny tne}m.

O`zwmng% qm wneb nic}w whm |ycinioeow cd whm yn`gck tnyoniem nzz}ko`a n tne}m qowho` n aotm`

o`wmytne.

O` cygmy wc gmwmyko`m whm |ycinioeow whnw n jc`wo`}c}z yn`gck tnyoniem nzz}kmz n tne}m o` n`

o`wmytne c} k}zw doyzw gynq whm d}`jwoc` d(x/.

\yc|mywomz cd n Jc`wo`}c}z Yn`gck Tnyoniem6

Whm |ycinioeow gm`zow d}`jwoc` d(x/ < dcy nee tne}mz cd x.

Whm |ycinioeow cd n jc`wo`}c}z yn`gck tnyoniem nzz}ko`a n tne}m qowho` zckm aotm` o`wmytne

dyck x> wc x9 oz gmdo`mg wc im whm nymn }`gmy whm ayn|h cd whm |ycinioeow gm`zow d}`jwoc`

imwqmm` x> n`g x9.

Whm |ycinioeow cd n jc`wo`}c}z yn`gck tnyoniem nzz}ko`a n z|mjodoj tne}m oz vmyc (whmym oz `cnymn }`gmy n` ayn|h nw n` mxnjw |co`w/.

Whm wcwne nymn }`gmy whm ayn|h cd d(x/ m~}nez >.

Mx|mjwmg Tne}m6

Whm mx|mjwmg tne}m (cy |c|}enwoc` kmn`/ cd n yn`gck tnyoniem o`gojnwmz owz ntmynam cy jm`wyne

tne}m. Ow oz n }zmd}e z}kkny tne}m (n `}kimy/ cd whm tnyoniem!z gozwyoi}woc`.

Zwnwo`a whm mx|mjwmg tne}m aotmz n am`myne ok|ymzzoc` cd whm imhntoc}y cd zckm yn`gck tnyoniemqowhc}w aoto`a d}ee gmwnoez cd owz |ycinioeow gozwyoi}woc` (od ow oz gozjymwm/ cy owz |ycinioeow gm`zow

d}`jwoc` (od ow oz jc`wo`}c}z/.

Wqc yn`gck tnyoniemz qowh whm znkm mx|mjwmg tne}m jn` hntm tmy goddmym`w gozwyoi}woc`z. Whmym

nym cwhmy }zmd}e gmzjyo|wotm kmnz}ymz qhojh nddmjw whm zhn|m cd whm gozwyoi}woc`% dcy mxnk|em

tnyon`jm.

8/8/2019 Discuss the various distributions of continuous random variable.

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Whm mx|mjwmg tne}m cd n yn`gck tnyoniem X oz zkiceozmg i M(X/ cy .

Od X oz n gozjymwm yn`gck tnyoniem qowh |czzoiem tne}mz x>% x9% x:% ...% x`% n`g |(xo/ gm`cwmz \(X ;

xo/% whm` whm mx|mjwmg tne}m cd X oz gmdo`mg i6

z}k cd xo.|(xo/

Qhmym whm memkm`wz nym z}kkmg ctmy nee tne}mz cd whm yn`gck tnyoniem X.

Od X oz n jc`wo`}c}z yn`gck tnyoniem qowh |ycinioeow gm`zow d}`jwoc` d(x/% whm` whm mx|mjwmg

tne}m cd X oz gmdo`mg i6

O`wmayne cd xd(x/gx

Mxnk|em6

Gozjymwm jnzm6 Qhm` n gom oz whycq`% mnjh cd whm |czzoiem dnjmz >% 9% :% 7% =% 4 (whm xo!z/ hnz n

|ycinioeow cd >#4 (whm |(xo/!z/ cd zhcqo`a. Whm mx|mjwmg tne}m cd whm dnjm zhcqo`a oz whmymdcym6

; M(X/ ; (> x >#4/ " (9 x >#4/ " (: x >#4/ " (7 x >#4/ " (= x >#4/ " (4 x >#4/ ; :.=

@cwojm whnw% o` whoz jnzm% M(X/ oz :.=% qhojh oz `cw n |czzoiem tne}m cd X.

Tnyon`jm6

Whm (|c|}enwoc`/ tnyon`jm cd n yn`gck tnyoniem oz n `c`+`manwotm `}kimy qhojh aotmz n` ogmn cd

hcq qogme z|ymng whm tne}mz cd whm yn`gck tnyoniem nym eobme wc im2 whm enyamy whm tnyon`jm% whmkcym zjnwwmymg whm cizmytnwoc`z c` ntmynam.

Zwnwo`a whm tnyon`jm aotmz n` ok|ymzzoc` cd hcq jeczme jc`jm`wynwmg yc}`g whm mx|mjwmg tne}m

whm gozwyoi}woc` oz2 ow oz n kmnz}ym cd whm !z|ymng! cd n gozwyoi}woc` nic}w owz ntmynam tne}m.

Tnyon`jm oz zkiceozmg i T(X/ cy Tny(X/ cy zoaknP9

Whm tnyon`jm cd whm yn`gck tnyoniem X oz gmdo`mg wc im6

T(X/ ; M(XP9/+M(X/P9

Qhmym% M(X/ oz whm mx|mjwmg tne}m cd whm yn`gck tnyoniem X.

@cwmz6

>. Whm enyamy whm tnyon`jm% whm d}ywhmy whnw o`gotog}ne tne}m cd whm yn`gck tnyoniem

(cizmytnwoc`z/ wm`gz wc im dyck whm kmn`% c` ntmynam2

8/8/2019 Discuss the various distributions of continuous random variable.

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9. Whm zkneemy whm tnyon`jm% whm jeczmy whnw o`gotog}ne tne}mz cd whm yn`gck tnyoniem

(cizmytnwoc`z/ wm`g wc im wc whm kmn`% c` ntmynam2

:. Wnbo`a whm z~}nym yccw cd whm tnyon`jm aotmz whm zwn`gnyg gmtonwoc`% o.m.6

Z~yw (T(X// ;zoakn

7. Whm tnyon`jm n`g zwn`gnyg gmtonwoc` cd n yn`gck tnyoniem nym neqnz `c`+`manwotm.

Z|mjone Yn`gck Tnyoniemz6

]`odcyk yn`gck tnyoniem.

@cykne yn`gck tnyoniem.

]`odcyk \ycinioeow Gozwyoi}woc`6

N yn`gck tnyoniem oz }`odcyke gozwyoi}wmg qhm`mtmy ow oz m~}nee eobme whnw n yn`gck tnyoniem jc}eg wnbm

c` n` tne}m imwqmm` j n`g g.

Whm }`odcyk yn`gck tnyoniem hnz whm dceecqo`a |ycinioeow gm`zow d}`jwoc`6

d(x/ ; >#(i + n/ dcy n 5 x 5 i

; < mezmqhmym

Qhmym% n ; zkneemzw tne}m whm tnyoniem jn` nzz}km

i ; enyamzw tne}m whm tnyoniem jn` nzz}km

Mx|mjwmg Tne}m cd x M(x/ ; (n " i/#9

Tnyon`jm cd x Tny(x/ ; (i + n/9#>9

@cykne \ycinioeow Gozwyoi}woc`6

Whm `cykne |ycinioeow gozwyoi}woc` oz whm kczw |c|}eny n`g ok|cywn`w gozwyoi}woc` dcy gmzjyoio`a n

jc`wo`}c}z yn`gck tnyoniem.

Whoz gozwyoi}woc` hnz imm` }zmg wc gmdo`m6

Whoz gozwyoi}woc` oz }zmg o` tnyoc}z zwnwozwojne o`dmym`jm wmjh`o~}mz

8/8/2019 Discuss the various distributions of continuous random variable.

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Hmoahwz n`g qmoahwz cd |mc|em

Wmzw zjcymz

O^ Zjcymz

Whm @cykne gozwyoi}woc` oz qogme }zmg o` tnyoc}z zwnwozwojne o`dmym`jm wmjh`o~}mz.Whm |ycinioeow gm`zow d}`jwoc` dcy n `cykne yn`gck tnyoniem oz6

Qhmym%k ; kmn`

z ; zwn`gnyg gmtonwoc`

| ; :.>7>=?

m ; 9.1>898

J}k}enwotm Gozwyoi}woc` D}`jwoc`6

Nee yn`gck tnyoniemz (gozjymwm n`g jc`wo`}c}z/ hntm n j}k}enwotm gozwyoi}woc` d}`jwoc`. Ow oz nd}`jwoc` aoto`a whm |ycinioeow whnw whm yn`gck tnyoniem X oz emzz whn` cy m~}ne wc x% dcy mtmytne}m x.

Dcyknee% whm j}k}enwotm gozwyoi}woc` d}`jwoc` D(x/ oz gmdo`mg wc im6

D(x/ ; \(X5;x/

dcy+o`do`ow 5 x 5 o`do`ow

Dcy n gozjymwm yn`gck tnyoniem% whm j}k}enwotm gozwyoi}woc` d}`jwoc` oz dc}`g i z}kko`a }| whm

|ycinioeowomz nz o` whm mxnk|em imecq.

Dcy n jc`wo`}c}z yn`gck tnyoniem% whm j}k}enwotm gozwyoi}woc` d}`jwoc` oz whm o`wmayne cd owz

|ycinioeow gm`zow d}`jwoc`.

Mxnk|emGozjymwm jnzm6 Z}||czm n yn`gck tnyoniem X hnz whm dceecqo`a |ycinioeow gozwyoi}woc` |(xo/6

xo < > 9 : 7 =

|(xo/ >#:9 =#:9 >#:9

Whoz oz njw}nee n io`ckone gozwyoi}woc`6 Io(=% #:9 4#:9 >4#:9 94#:9 :>#:9 :9#:9

D(x/ gcmz `cw jhn`am nw o`wmykmgonwm tne}mz. Dcy mxnk|em6D(>.:/ ; D(>/ ; 4#:9

D(9.84/ ; D(9/ ; >4#:9

\ycinioeow Gm`zow D}`jwoc`6

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Whm |ycinioeow gm`zow d}`jwoc` cd n jc`wo`}c}z yn`gck tnyoniem oz n d}`jwoc` qhojh jn` im

o`wmaynwmg wc ciwno` whm |ycinioeow whnw whm yn`gck tnyoniem wnbmz n tne}m o` n aotm` o`wmytne.

Kcym dcyknee% whm |ycinioeow gm`zow d}`jwoc`% d(x/% cd n jc`wo`}c}z yn`gck tnyoniem X oz whm

gmyotnwotm cd whm j}k}enwotm gozwyoi}woc` d}`jwoc` D(x/6 d(x/ ; g#gx D(x/

Zo`jm D(x/ ; \(X5;x/ ow dceecqz whnw6 o`wmayne cd d(x/gx ; D(i/+D(n/ ; \(n5X5i/

Od d(x/ oz n |ycinioeow gm`zow d}`jwoc` whm` ow k}zw cim wqc jc`gowoc`z6

>. Whnw whm wcwne |ycinioeow dcy nee |czzoiem tne}mz cd whm jc`wo`}c}z yn`gck tnyoniem X oz >6

O`wmayne cd d(x/gx ; >

9. Whnw whm |ycinioeow gm`zow d}`jwoc` jn` `mtmy im `manwotm6 d(x/ 0 < dcy nee x.

Gozjymwm Yn`gck Tnyoniem6

N gozjymwm yn`gck tnyoniem oz c`m qhojh kn wnbm c` c`e n jc}`wniem `}kimy cd gozwo`jw tne}mz

z}jh nz % 9% :% 7% ... Gozjymwm yn`gck tnyoniemz nym }z}nee (i}w `cw `mjmzznyoe/ jc}`wz. Od n

yn`gck tnyoniem jn` wnbm c`e n do`owm `}kimy cd gozwo`jw tne}mz% whm` ow k}zw im gozjymwm.Mxnk|emz cd gozjymwm yn`gck tnyoniemz o`je}gm whm `}kimy cd jhoegym` o` n dnkoe% whm Dyogn

`oahw nwwm`gn`jm nw n jo`mkn% whm `}kimy cd |nwom`wz o` n gcjwcy!z z}yamy% whm `}kimy cd

gmdmjwotm eoahw i}eiz o` n icx cd wm`.

O`gm|m`gm`w Yn`gck Tnyoniemz6

Wqc yn`gck tnyoniemz X n`g _ zn% nym znog wc im o`gm|m`gm`w od n`g c`e od whm tne}m cd X hnz

`c o`de}m`jm c` whm tne}m cd _ n`g tojm tmyzn. Whm j}k}enwotm gozwyoi}woc` d}`jwoc`z cd wqc

o`gm|m`gm`w yn`gck tnyoniemz X n`g _ nym ymenwmg iD(x%/ ; A(x/.H(/

Qhmym%

A(x/ n`g H(/ nym whm knyao`ne gozwyoi}woc` d}`jwoc`z cd X n`g _ dcy nee |noyz (x%/.

B`cqemgam cd whm tne}m cd X gcmz `cw mddmjw whm |ycinioeow gozwyoi}woc` cd _ n`g tojm tmyzn.Wh}z whmym oz `c ymenwoc`zho| imwqmm` whm tne}mz cd o`gm|m`gm`w yn`gck tnyoniemz.

Dcy jc`wo`}c}z o`gm|m`gm`w yn`gck tnyoniemz% whmoy |ycinioeow gm`zow d}`jwoc`z nym ymenwmg i

d(x%/ ; a(x/.h(/

Qhmym%a(x/ n`g h(/ nym whm knyao`ne gm`zow d}`jwoc`z cd whm yn`gck tnyoniemz X n`g _ ymz|mjwotme%

dcy nee |noyz (x%/.

Dcy gozjymwm o`gm|m`gm`w yn`gck tnyoniemz% whmoy |ycinioeowomz nym ymenwmg i

\(X ; xo 2 _ ; f/ ; \(X ; xo/.\(_;f/

dcy mnjh |noy (xo%f/.

8/8/2019 Discuss the various distributions of continuous random variable.

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YMDMYM@JMZ6

Yn`gck Kmnz}ymz% 7wh mgowoc`

Dc}`gnwoc`z cd Kcgmy` \ycinioeow% 9`g mgowoc`.

hww|6##qqq.zwnwz.aen.nj.}b#zwm|z#aeczzny#|ycinioeowSgozwyoi}woc`z.hwke. \ycinioeow% Yn`gck Tnyoniemz% n`g Zwcjhnzwoj \ycjmzzmz. KjAynqHoee Bcanb}zhn.

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