2
INTRODUCTION
• Random Variable: I i! a "ariable #$i%$ a&e! !'e%i(ied "al)e! #i$
!'e%i(ied 'robabiliie!*
• In o$er #ord!+ i i! a real n)mber %onne%ed #i$ $e o)%ome o( a
random e,'erimen*
• I %an be o( #o -'e!: Di!%ree and Conin)o)!*
• Probabili- Di!rib)ion: Li!in. o( all $e 'o!!ible o)%ome! o( a
random "ariable #i$ ea%$ o)%ome/! a!!o%iaed 'robabili- o(
o%%)rren%e i! %alled 'robabili- di!rib)ion*
 
BINOMIAL DISTRIBUTION
• I i! a #idel- )!ed 'robabili- di!rib)ion (or a di!%ree random
"ariable*
• T$e di!rib)ion i! al!o &no#n a! $e o)%ome o( a Berno)lli 'ro%e!!
and i! a!!o%iaed #i$ $e name o( 0a%ob Berno)lli*
• A 'ro%e!! in #$i%$ ea%$ rial $a! onl- #o 'o!!ible o)%ome!+ $e
 'robabili- o( $e o)%ome a an- rial remain (i,ed o"er ime+ and
$e rial! are !ai!i%all- inde'enden
 
CONDITIONS 1OR 
 BINOMIAL E2PERIMENT 3* T$e random e,'erimen i! 'er(ormed )nder $e !ame %ondiion!
(or a (i,ed and (inie n)mber o( rial!+ !a- n*
4* Ea%$ rial i! inde'enden o( o$er rial!+ i*e*+ $e 'robabili- o( an
o)%ome (or an- 'ari%)lar rial i! no in(l)en%ed b- $e o)%ome!
o( $e o$er rial!*
5* Ea%$ rial $a! #o m))all- e,%l)!i"e 'o!!ible o)%ome!* T$e
o)%ome! are )!)all- %alled !)%%e!! and (ail)re*
* T$e 'robabili- o( !)%%e!! +'+ remain! %on!an (rom rial o rial
7!o i! $e 'robabili- o( (ail)re 8+ #$ere 893'; *
 
1UNCTION
• 1or a binomial random "ariable+ 'robabili- o( obainin. e,a%l- ,
( )
*>>
*
where
 x
n
• T$e mean o( binomial random "ariable 2+ i! .i"en a!:
 Mean9 n'
• T$e "arian%e o( $e binomial random "ariable 2 mea!)re! $e "ariaion o( $e binomial di!rib)ion and i! .i"en b-:
  V72;9 n'8
POISSON DISTRIBUTION
• T$e di!rib)ion i! named a(er $e 1ren%$ ma$emai%ian Simeon
Poi!!on*
• T$e Poi!!on di!rib)ion o%%)r! #$en $ere are e"en! #$i%$ do no
o%%)r a! o)%ome! o( a de(inie n)mber o( rial! o( an e,'erimen+
 b) #$i%$ o%%)r a random 'oin! o( ime and !'a%e+ #$ere o)r
inere!! lie! onl- in $e n)mber o( o%%)rren%e! o( $e e"en! and no
in i! nono%%)rren%e!*
• T$e n)mber 2 o( o)%ome! o%%)rrin. d)rin. a 'oi!!on e,'erimen
i! %alled a! a 'oi!!on random "ariable+ and $e di!rib)ion $a i
(ollo#! i! ermed a! $e Poi!!on di!rib)ion*
 
C@ARACTERSTICS PROPERTIES
• A Poi!!on e,'erimen i! deri"ed (rom $e Poi!!on 'ro%e!! #$i%$ $a!
$e (ollo#in. 'ro'erie!%$ara%eri!i%!:
3* T$e n)mber o( o)%ome! o%%)rrin. in one iner"al i! inde'enden o(
$e n)mber $a o%%)r! in !ame or an- o$er di!oin ime iner"al*
4* An in(inie n)mber o( o%%)rren%e! o( $e e"en m)! be 'o!!ible in
$e iner"al*
5* In an- e,remel- !$or 'orion o( iner"al+ 'robabili- o( #o or
more o%%)rren%e! o( $e e"en i! ne.li.ible*
 
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DI11ERENCE
• Poi!!on di!rib)ion di((er! (rom $e Binomial di!rib)ion in #o
im'oran a!'e%!:
3* Ra$er $an %on!i!in. o( (i,ed n)mber o( rial!+ $e di!rib)ion
o'erae! %onin)o)!l- o"er !ome .i"en amo)n o( ime+ di!an%e+
area e%*
4* Ra$er $an 'rod)%in. a !e8)en%e o( !)%%e!!e! (ail)re!+ $e
di!rib)ion onl- (o%)!e! on !)%%e!!e!+ #$i%$ are &no#n a! $e
<o%%)rren%e!/*
POISSON DISTRIBUTION
• A random "ariable 2 i! !aid o (ollo# a Poi!!on di!rib)ion i( i
a!!)me! onl- nonne.ai"e "al)e!+ and i! .i"en b-:
• T$e Poi!!on di!rib)ion i! a limiin. (orm o( Binomial di!rib)ion
#$en <'/ i! $e %on!an 'robabili- o( !)%%e!! (or ea%$ rial i! "er-
!mall and <n/ $e n)mber o( rial! i! inde(iniel- lar.e+ !)%$ $a
n'9
*+***+4+3+?;7   n xwhere  x
e  X  P 
 NORMAL DISTRIBUTION
• I i! $e mo! im'oran %onin)o)! di!rib)ion+ and #a! di!%o"ered
 b- De Moi"re*
• T$e normal di!rib)ion i! al!o &no#n a! $e a)!!ian Di!rib)ion*
•  Normal di!rib)ion i! a limiin. %a!e o( binomial di!rib)ion*
• $en <n/ i! "er- lar.e and nei$er <'/ nor <8/ i! "er- !mall+ binomial
di!rib)ion end! o normal di!rib)ion*
• Binomial di!rib)ion end! o $e (orm o( $e %onin)o)! %)r"e
#$en <n/ be%ome! lar.e*
 
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 NORMAL DISTRIBUTION • Probabili- ()n%ion o( $e Normal di!rib)ion i! .i"en a!:
F3G5*4an
withondistributi Normal   followswhichiablerandomais X 
PROPERTIES O1 NORMAL CURVE
3* T$e %)r"e i! bell!$a'ed and !-mmeri%al abo) mean*
4* T$e mean+ median and mode o( $e normal di!rib)ion %oin%ide*
5* T$e oal area )nder $e %)r"e and abo"e $e $oriJonal a,i! i! e8)al o )ni-*
* Area )nder $e %)r"e on $e le( o( $e mean i! e8)al o $e area )nder %)r"e o $e ri.$ o( $e mean*
K* T$e normal di!rib)ion i! !-mmeri%al and Me!o&)ri%+ i*e*+ 39?
and 495*
 
STANDARD NORMAL VARIATE
• A random "ariable #i$ an- mean and S*D* %an be ran!(ormed o a
!andardiJed normal "ariable b- !)bra%in. $e mean and di"idin.
 b- S*D*
• 1or a normal di!rib)ion #i$ mean and S*D* + $e !andardiJed
"ariable J i! .i"en a!:
• T$e !andardiJed "ariable <J/ i! %alled a !andard normal "ariae+
#$i%$ (ollo#! Normal di!rib)ion #i$ mean ? and S*D* 3+ i*e*+
J H N7?+ 3;