lecture 14

UNR, MATH/STAT 352, Spring 2007

Binomial(n,p)

!( ) (1 ) (1 )

( )! !k n k k n kn n

P X k p p p pk n k k

( )E X np 2 (1 )X np p


0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

Number of succsses

db

ino

m(q

, 1, 0

.5)

Binomial(1,0.5)

Number of successes within 1symmetric Bernoulli trialcan only be 0 or 1. These

possibilities have equal chances.


0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

Number of succsses

db

ino

m(q

, 2, 0

.5)

Binomial(2,0.5)

Number of successes within 2symmetric Bernoulli trials

can only be 0, 1 or 2. The possibility to have exactly 1 success

is larger than that of having 0 or 2.

A fair game should result in a tie.

Then why do people play fair games?


0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

Number of succsses

db

ino

m(q

, 2, 0

.1)

Binomial(2,0.1)

Most likely there will be no successes


0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

Number of succsses

db

ino

m(q

, 2, 0

.9)

Binomial(2,0.9)

Most likely there will be only successes


0 2 4 6 8 10

0.0

00

.05

0.1

00

.15

0.2

00

.25

0.3

00

.35

Number of succsses

db

ino

m(q

, 15

, 0.1

)

Binomial(15,0.1)

Unimodal (mode = 1)

Right-skewed

Concentrated around 1.5

(E = np = 1.5)


0 2 4 6 8 10

0.0

00

.05

0.1

00

.15

0.2

00

.25

Number of succsses

db

ino

m(q

, 19

, 0.1

)

Binomial(19,0.1)

Unimodal (mode = 1-2)

Right-skewed

Concentrated around 1.5


0 5 10 15 20 25 30

0.0

00

.02

0.0

40

.06

0.0

80

.10

0.1

2

Number of succsses

db

ino

m(q

, 10

0, 0

.1)

Binomial(100,0.1)

Unimodal (mode = 10)

Symmetric? (E = np = 10)

Concentrated around 10

P(10+3) < P(10-3)


70 75 80 85 90 95 100

0.0

00

.02

0.0

40

.06

0.0

80

.10

0.1

2

Number of succsses

db

ino

m(q

, 10

0, 0

.9)

Binomial(100,0.9)

Unimodal (mode = 90)

Symmetric? (E = np = 90)

Concentrated around 90


0 20 40 60 80 100

0.0

00

.02

0.0

40

.06

0.0

80

.10

0.1

2

Number of succsses

db

ino

m(q

, 10

0, 0

.1)

Bin(100,0.9)Bin(100,0.1)

Only a small fraction of possible outcomes has not negligible P (i.e. only small part can be seen

in experiment)

P is very small (not 0!) here


0 1 2 3 4 5 6

0.0

00

.05

0.1

00

.15

0.2

00

.25

0.3

0

Number of succsses

db

ino

m(q

, 6, 0

.5)

Binomial(6,0.5)

Here all possibleoutcomes have reasonable

probabilities


Poisson()

( )!

k

P X k ek

( )E X 2X


0 2 4 6 8 10

0.0

0.1

0.2

0.3

Number of succsses

dp

ois

(q, 1

)Poisson(1)

0 2 4 6 8 10

0.0

0.1

0.2

0.3

Number of succsses

db

ino

m(q

, 10

00

, 1/1

00

0)

Binomial(1000,.001]

I’ve seen this already!


n np 0p+ +

Binomial(n,p) Poisson()UNR, MATH/STAT 352, Spring 2007

0 10 20 30 40 50

0.0

00

.02

0.0

40

.06

Number of succsses

dp

ois

(q, 3

0)

0 10 20 30 40 50

0.0

00

.02

0.0

40

.06

Number of succsses

dp

ois

(q, 3

0)

0 10 20 30 40 50

0.0

00

.02

0.0

40

.06

Number of succsses

dp

ois

(q, 3

0)

Poisson

Binomial

Normal

Poisson(30) Binomial(1000,.03) N(30,sqrt(30))


If n is large (n > 100), p is small (p < 0.05), andboth np and n(1-p) are not small (say >10) then

B(n,p)~P(np)~N(np, np(1-p))


Normal densities

Value of random variable

De

nsi

ty

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mean= 7, Std=1

Mean= 5, Std=1Mean= 0, Std=1

Mean= 0, Std=0.6

Mean= 0, Std=3


2400 2450 2500 2550 2600

0.00

00.

006

0.01

2Probability density function


Pro

babi

lity

Lower-tail probability

Interval probability

Upper-tail probability

2400 2450 2500 2550 2600

0.0

0.4

0.8

Cumulative distribution function


Pro

babi

lity



12Payoff 2 , where Bin( , )

(Why this formula? See lecture notes.)

X N X N

-100 -50 0 50 100

0.0

0.1

0.2

0.3

0.4

0.5

2 games. P(lose or win < $10)= 1

Net payoff, in $

Pro

babi

lity

of p

ayof

f




X N X N

-100 -50 0 50 100

0.0

0.1

0.2

0.3

0.4

0.5


Net payoff, in $

Pro

babi

lity

of p

ayof

f




X N X N

-100 -50 0 50 100

0.0

0.1

0.2

0.3

0.4

0.5

100 games. P(lose or win < $10)= 0.728

Net payoff, in $

Pro

babi

lity

of p

ayof

f




X N X N

-100 -50 0 50 100

0.0

0.1

0.2

0.3

0.4

0.5


Net payoff, in $

Pro

babi

lity

of p

ayof

f

lecture 14

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