Special Distributions

Bernoulli and Binomial distributions

Ipmf, mean, variance, mgf.

Xv Bernoulli ( p) if Xel 0,13 Pr(x=D=p

ECX )=p Var(x)=E(X4 . Elx5=p-p2=p( 1- p)

4H7=E(qt×)=l- ftp.et.p.tp.p.et

a 9

Xrr Binomial ( w ,p )X=# Of Is in n Bernoulli trials

X=X,tXzt . . .tXw Xi , it , ... ,w iid Bernoulli (p )

ECX )=np Var(X)=Var1k)t . . . .+Var( xD ( independence)= wpctp )

YH )=E(et×)=E(etkt.it#)=E(etx'.etx2.....etxy=E(etxD.E(etxY.....E(et*) ( independence )

=( iptpet ) ftp.ipet ) . :( tptpet)= ( i - ptpet )

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Example

Three men A, B, and C shoot at a target. Suppose that A shoots three

times and the probability that he will hit the target on any given shot is

1/8, B shoots five times and the probability that he will hit the target onany given shot is 1/4, and C shoots twice and the probability that he willhit the target on any given shot is 1/2. What is the expected number oftimes that the target will be hit?

E ( # of hits from A) = 3 . to

E (# of hits from B) = 5 . ty

E ( # of hits from c) = 2 . t

E ( total # of hits ) = of + 5g + ÷ = . . .

Hypergeometric distribution

Suppose that a box contains A red balls and B blue balls. Suppose also

that n � 0 balls are selected at random from the box withoutreplacement, and let X denote the number of red balls that are obtained.

Find the pmf of X .

box contains ATB balls

#e⇒Pr(x=x)=(n±⇒

he At B

OE K c- A

OE h - x e 13

Hypergeometric distribution

Let A, B , and n be nonnegative integers with n A+ B . If a randomvariable X has a discrete distribution with pmf as in the example above,

then it is said that X has the hypergeometric distribution with

parameters A, B , and n.

Find the mean and the variance of a hypergeometric distribution.

Xi ELO , 1) X, .=1 if the ilth ball is red

X=X , txzt . . .+×w

Nole that Xi , Xz , . . . . , Xu are NOT independent!

k€20,1 ] Pr( X ,=D=Pr( first ball is red ) = It X ,~Bern°uHi(p=a÷3 )

Xztlai } Pr(Xz=D=PrFXz=l ,X,=o)+Pr( X .=l , 4=1 )=PrCx,=D .Pr(×z=i1K=o)tPr(x,=D . Pr( Xitlx ,=i )= '¥, . ¥ta÷¥

AC Bta - l )=±

= (A#t3 At 13

Xz~ Bernoulli ( p=n÷,3 ) ( the same as X , )

Ktla ' ) Pr(X}=D= AMI X ,~ Bernoulli ( p - ate, )

Xi ~ Bernoulli ( p=a÷y )

X = X , tXz+ . . . +Xw X., N Bernoulli (p= IB ) ( not indep)

E( D= ? varlx ) = ?

Elx )=E(×Dt . ..tt#=YntT3Var(X)=Var(kt...+Xn)=Var(xDtVarCxD+...tVar(

xD + 2 . EIZ w(×is× ;)

= HET +24¥ uucx , ,x . )= nμ?pX,y + Mm ) Car ( x , ,×D = - (*kf÷t3J

ark ,xD=EKx . ) - EADEHD ; !gaj÷ -

t.I.ae#IEE*3EtE**x,kelqHPr(kX.=D=PrCX.=l, ×a= ' ) =Pr(x , =D . Pr(k=l/x,=i )

=n÷a . aA¥'