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3.1

Random

Variables

and

Discrete

DistributionsA

random

variable

is

some

function

thatassigns

a

real

number

X(s)to

each

possiblee

s

S

,

where

S

is

the

sample

space

foran

experiment.Discrete:e.g.Female=1,

Male=0e.g.

Satisfied=1,Average=0,

Unsatisfied=-1Continuous:

height,

score,

priceDiscrete

DistributionsExample:

Coke

vs

PepsiTen

people

are

surveyed

about

whether

theypreference

Coke

or

Pepsi. The

sample

spacecan

be

regarded

as

the

set

of

210

differentsequences.X

could

bethe

number

of

people

who

preferCoke.PPCPPPCPPPCPPPPCP……X(s)=4Example:

Suppose

the

Environment

ProtectionAgency

(EPA)

takes

readings

once

a

month

onthe

amount

of

pesticide

in

the

discharge

water

ofa

chemical

company.

If

the

amount

of

pesticideexceeds

the um

level

set

by

the

EPA,

thecompany

is

forced

to

take

corrective

action

andmay

be

subject

to

penalty.Consider

the

random

variable,

X,

that

is

thenumber

of

months

until

the

company’s

dischargeexceeds

the

EPA’s um

level.What

values

can

X

assume?Solution:The

company’s

discharge

of

pesticide

mayexceed

theum

allowable

level

on

the monthoftesting,the

second

month

of

testing,

and

so

on.It

is

possible

that

the

company’sdischarge

will

neverexceed

the um

level.Thus,the

set

of

possible

values

for

the

number

ofexceeded

is

the

set

ofallmonthsuntil

the

level

ispositive

integers1,2,3,4,…隨量（random

variable)：是實驗結果的數(shù)值描述。試驗隨

量隨 量的可能值與5名客戶聯(lián)系訂貨的客戶數(shù)量0，1，2，3，4，5檢查裝運的50臺收音機次品收音機的數(shù)量0，1，2，…,49,

50飯店營業(yè)一天客戶的數(shù)量0，1，2，3，…銷售一輛汽車客戶的如果是

則為0；如果是女性則為1。表1：離散型隨 量舉例經(jīng)營一家銀行裝灌一種軟飲料（最大值=12.1盎司）盎司數(shù)x>=00<=x<=12.1試驗隨

量

隨客戶在銀行停留的分鐘數(shù)量的可能值表2：連續(xù)型隨量舉例A

complete

description

of

a

discrete

randomvariable

requires

that

we

specifythe

possible

values

the

random

variable

can

assumethe

probability

associated

witheach

value.ProblemSuppose

that

in

some

population,

the

probability

of

preferring

Coke

over

Pepsi

is

50%.Two

people

are

surveyed.Let

X

be

the

number

of

people

who

prefer

Coke.Find

the

probability

associated

with

each

valueof

the

random

variable

X.

Display

these

valuesin

a

table

raph.Solution:P(

X

0)

P(PP)

1

/

4P(

X

1)

P(PC)

P(CP)

1/

2P(

X

2)

P(CC)

1

/

41/41/2f

(x)0

1

2xNow

we

know

the

values

therandom

variable

can

assume(0,1,2)

and

how

the

probability

isdistributed

over

these

values(1/4,1/2,1/4).

This

comple

ydescribes

the

distribution

of

therandom

variable

and

is

referredto

as

the

probability

function,denoted

by

the

notation

f(x).Discrete

DistributionsX

has

a

discrete

distribution

if

X

can

take

only

afinite

number

of

different

values

x1,...,xk

or,atmost,

an

infinite

sequence

of

different

valuesx1,x2,...The

probability

function

(p.f.)

of

X

is

definedtobe

the

function

fsuch

that

for

every

realnumber

x,

f(x)=Pr(X=x).If

x

is

not

one

of

the

possible

values

of

X,f(x)=0.We

haveorki1if

(x

)

1i1If

A

is

any

subset

ofthe

real

line,Pr(

X

A)

f

(xi

)xi

Aif

(x

)

1The

Distribution

FunctionThe

distribution

function

of

a

random

variableX

is

a

function

defined

for

each

real

numberx:

Since

F(x)

is

the

probability

of

the

event

{X

x}At

any

point

x,0

F

(x)

1Thed.f.

ofa

Discrete

DistributionxxIf

a<b

and

if

Pr(a<X<b)=0,

then

F(x)will

beconstantand

horizontal

over

the

interval

a<X<b.At

every

point

x

such

that

Pr(X=x)>0,

the

d.f.

willjump

by

the

amountPr(X=x).f(x)

F(x)The

BinomialDistributionE.g.

A

machine

produces

a

defective

item

withprobability

p

(0<p<1)

and

produces

anondefectiveitem

with

probability

q=1-p.n

independent

itemsare

examined. Let

X

denote

thenumber

of

these

items

that

are

defective.

X

can

takevalues

0,1,2,...,n.The

BinomialDistributionThis

discrete

distribution

is

called

Binomialdistribution

with

parameters

n

and

p.

x

nfor

x

0,1,...notherwisex n

x0f

(x)

p

qfor x

0for j

x

j

1,

j

0,1,,

(n

1)jpiqnii0for x

n

n

F

(x)

i0

1

Continuous

DistributionsXhas

a

continuous

distribution

ifthere

exists

anonnegative

function

f

defined

on

the

realline,

s.t.

forany

subset

A

of

the

real

line,Pr(

X

A)

f

(x)dxAf

is

called

the

probability

density

function(p.d.f.)

of

X. Every

p.d.f.

must

satisfy

tworequirements:f

(x)

0

f

(x)dx

1The

probability

density

function

(pdf)

for

acontinuous

random

variable

is

afunction

which

canbe

integrated

toobtain

the

probability

that

the

randomvariable

takes

a

value

ina

given

interval.P(a<=X<=b)=P(X=a)=0f

(x)dxbaContinuous

DistributionsProbability

for

IndividualValuesSoPr(a

X

b)

Pr(a

X

b)

Pr(a

X

b)

Pr(a

X

b)For

every

individual

value

x,xxf

(t)dt

0Pr(

X

x)

The

d.f.

of

a

Continuous

DistributionSuppose

X

has

a

continuous

distribution

with

p.d.f.f(x). Since

the

probability

of

each

individual

point

xis

0,

the

d.f.

F(x)

willhave

no

jumps.Furthermore,xF

( )

f

(t)dtdxat

each

point

x

at

which

f(x)

is

continuous,F

'(x)

dF

(x)

f

(x)Uniform

Distribution

on

An

Intervala

and

b

are

two

givenreal

numbers

suchthata<b.A

point

X

is

selected

from

the

intervalS

{x

:

a

x

b}The

probability

that

X

will

belong

to

anysubinterval

of

S

is

proportional

to

the

lengthof

that

subinterval.This

distribution

is

called

the

uniformdistribution

on

the

interval

(a,b).The

p.d.f.f(x)

of

Xmust

be

0

outside

S.f(x)

must

be

constant

throughout

S.Also,The

p.d.f.

of

Xmust

beba

Sf

(x)dx

f

(x)dx

1for

a

x

botherwise1f

(x)

b

a

0b

afor x

afor

a

x

bfor x

b

10F

(x)

x

aThe

value

of

the

constantis

the

reciprocal

of

thelength

oftheinterval.It

is

not

possible

to

define

a

uniform

distributionoverthe

interval

x

a because

the

length

of

this

intervalis

infinite.Since

the

probability

is

0

at points

a

or

b,

itisirrelevant

whetherthe

distribution

is

regarded

as

auniform

distribution

on

[a,b],

(a,b],

[a,b)

or

(a,b).Example Calculating

Probabilities

from

ap.d.f.0

otherwiseThep.d.f.

of

a

X

has

the

followingform:for

0

x

4f

(x)

cxc

?Pr(1

X

2)

?Pr(X

2)

?Example Calculating

Probabilities

from

ap.d.f.Solution:0

otherwiseThep.d.f.

of

a

X

has

the

followingform:for

0

x

4f

(x)

cxc

?Pr(1

X

2)

?Pr(X

2)

?14042

8Pr(X

2)

1

xdx

34163xdx

Pr(1

X

2)

cxdx

8c

1

c

182

18Note:

c

is

often

called

the

normalizing

constant.Its

value

is

unique.102

116for x

0x for

0

x

4for x

4F

(x)

Example:

Unbounded

Random

Variables(1

x)2for

x

01

0In

an

electrical

system,

the

voltage

X

is