Small Mathematical Expectation

8/17/2019 Small Mathematical Expectation

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1

Mathematical Expectation

of a Random Variable

2

Idea: mean value of a random

variable

• Definition: Weighted mean of the values ofthe random variable

• Condition for the existence:

– Note: Infinite sets can yield paradoxes

∑∈

== X x

x X xP X E )()(

∞


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7

Idea: mean value of a random

variable• Note that it is probab i l i s t i c in te rp re tat ion of the

common sense concept of mean .

• Note that if all possible samples are equallyprobable, they are equivalent.

∑∈= X x x N X Mean1

)(

1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 2 2 00

5

10

15

20

25

1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 2 2 0 2 3 00

10

20

30

40

50

60

70

80

90

1001 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 00

5

10

15

20

25

1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 00

10

20

30

40

50

60

70

80

90

10 0

X X

∑∈ == X x x X xP X E )()( N x X P 1

)( ==

8

Idea: center of mass/equilibrium

• If the probability P(X =x) is interpreted asmass, and the random variable X asdistance, the mathematical expectation isthe center of mass of the object.

∑∈

== X x

x X xP X E )()(

9

Example: Roulette

• What is the payoff given by the casino?• Roulette selects at random a number between 1 and 36 (0) .

• Players can bet on 18,12,9,6,4,3,2,1 number.

– A Bet over k numbers has a probability of success of k/36 andof getting a payment x from the casino or lossing a.

– Expected value is

– A fair game would imply

( )ak

xk

x X xP X E LossWin x

−

−+=== ∑

∈ 361

36)()(

},{

ak

x

X E

−=

=

136

0)(

10

Example: Roulette

• What is the payoff given by the casino?

• The roulette has 37 slots (1 to 36+ the 0 slot) – In a Real game the casino the odds are

• 35:1 17:2 11:3

( )3737

137

136

)()(},{

ak a

k a

k x X xP X E

LossWin x

−=

−−+

−=== ∑

∈

0(


3/6

13

Expected value of a geometric

random variable

• Random Varible X={Number of trials untila success}

• How do we sum this series?

L1,2,3,ifor)1()( 1 =−== − p pi X P i

( )L++++−=−= ∑

∞

=

− 32

0

1

4321)1()1()( p p p p pnp X E n

n

14


random variable


15


random variable


16


random variable


( )

( )

( )

1)1(

1)1()1(

1)1()()(

432)1()1()(

4321)1()1()(

Series

Geometric

0

432

432

1

32

0

1

=

−

−=−=

=+++++−=−

++++−=−=

++++−=−=

∑

∑

∑

∞

=

∞

=

∞

=

−

p p p p

p p p p p X pE X E

p p p p p pnp X pE

p p p p pnp X E

k

k

n

n

n

n

L

L

L

)1(

1 )(

p X E

−=

17


random variable• How do we sum this series? Another way

2

21

1

Series Geometric

1

)(

)()()()(

)(

)(

)1(

1

)1()(

)1()(

xb

xbdx

d xa xa

dx

d xb

xb

xa

dx

d

p p

p

dp

d kp pS

dp

d

p

p p pS

k

k

k

k

−=

−=

−

==

−==

∑

∑∞

=

−

∞

=

2)1(

1)1()(

p p X E

−−=

18


random variable• Random Varible X={Number of trials until

a success}

L1,2,3,ifor)1()( 1 =−== − p pi X P i

( )

p X E

p p p p pnp X E n

n

−=

++++−=−=∑∞

=

−

1

1)(

4321)1()1()(32

0

1L

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 6 1 1 1 6 2 1 2 6 3 1 3 6 4 1 4 6 5 1 5 6 6 1

9

1

1

1)( =

−=

p X E


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19

Relation of expectaition and other

statistical measures

• Skew distribution vs. symetric distribution

Full House: The Spread of Excellence from

Plato to Darwin by Stephen Jay Gould

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

-8 -6 -4 -2 0 2 4 6 80

0.2

0.4

0.6

0.8

Mode Median MeanMode Median Mean

20

Property of lineality

• The expectation of a linear combination of randomvariables is the linear combination of expectations.

( ) ( )( )

( ) ( )

)()(

)()(

)()(

)()()(

Y E X E

P Y P X

P Y X Y X E

Y E X E Y X E

βα

ωωβωωα

ωωβωαβα

βαβα

ωω

ω

+=

=+=

+=+

+=+

∑∑

∑

Ω∈Ω∈

Ω∈

21

Finding the maximum

• Suppose that n children of differing heights areplaced in line at random. You select the first

child from the line and walk with her/him alongthe line until you encounter a child who is talleror until you have reached the end of the line. If

you do encounter a taller child, you repeat theprocess.

• What is the expected value of the number of

children selected from the line?

Taken fromTijms, Understanding probability22

Finding the maximum

• We define the variable X as the number ofchildren selected from the line.

• We will define the indicator variable:

• Now the number of selected children will

be:

= otherwise 0

linethefromselectedischildtheif 1 i-th X i

n X X X X +++= L21

23

Finding the maximum

• The probability that the ith child is thetallest among the first i children is 1/i

• Therefore:

( ) n1,2,ifor11

11

10 L==⋅+

−⋅=

iii X E i

( ) ( )

( ) 57722.02

1)ln(

1

3

1

2

11

21

++≈+++=

+++=

nn

n X E

X X X E X E n

L

L

24

Expected number of distinct

birthdays• What is the expected number of distinct

birthdays within a randomly formed groupof 100 persons.

– We define the random variable

– The number of birthdays is

= otherwise 0

dayonis birthdaytheif 1 i X i

36 521 X X X X +++= L

Taken fromTijms, Understanding probability


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birthdays

• What is the expected number of distinct birthdays withina randomly formed group of 100 persons.

– For each day we have:

– The expected number of distinct birthdays is

)0(1)1(

365

364)0(

100

=−==

==

ii

i

X P X P

X P

Taken fromTijms, Understanding probability

( ) ( ) 6.87365

3641365

10 0

36 521 =

−=+++= X X X E X E L

26


birthdays

• For an arbitrary number of persons.

( )

−=

n

X E 365

3641365

50 100 150 200 250 300 350

50

100

150

200

250

300

350

Number of persons

E x p e c t e d n u m b e r o f d i s t i n c t b i r t h d a y s

25 30 35 40 45

20

25

30

35

40

45

50

27

Other Properties

• If X is a non negative random variable,then

• Also

0)( ≥ X E

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 6 1 1

1 6

2 1

2 6

3 1

3 6

4 1

4 6

5 1

5 6

6 1

)()( then Y E X E Y X ≥≥

28

Other Properties

• What do we mean by

?)()( then Y E X E Y X ≥≥

ℜ→Ω: X

0 1 X(.) X(.)

Y(.) Y(.)

X

0 1 P(X) P(X)

P(Y) P(Y)

X

ℜ→Ω:Y

X

∑∈

== X x

x X xP X E )()(

29

Caveats of intuition

• Does the expectation always exist?

∑∈

= X x

x N

X Mean1

)(

∑∈

== X x

x X xP X E )()(

http://physicsweb.org/articles/world/14/7/9/1#pw1407091

The physics of the WebJuly 2001

λλ −== e x

x X P x

!)(

α x

x X P 1

)( ==

Sample mean always exists

Expectation

perhaps givesan infinite

value !!!!

30


• Does the expectation always exist?

• Example: A Cauchy random variable

∑∈

= X x

x N

X Mean1

)(

∑∈

== X x

x X xP X E )()(

http://physicsweb.org/articles/world/14/7/9/1#pw1407091

The physics of the WebJuly 2001

( ) ( )+∞


6/6

31


• Why is infinite?

• Divergent series

• Note that for high values of x

( )∑∑ ∞

=∈ +===

1

21

)()( x X x x

x x X xP X E

π

α x x X P

1)( ==

( ) ( )

∑

∑ ∑∑∞

=

∞

=

∞

>>

>>

=

∝

++

≅+

=

1

1 1

1

1

22

1)(

1111

)(2

2

x

x x

x

x

x X E

x x x

x x X E

πππ

32


• Why is infinite?

• Value of the harmonic series

0 10 00 200 0 3 00 0 4 000 5 000 6 000 7 00 0 8 00 0 900 0 100005

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

∑=

N

x x1

1

∞=+++>

++++++++>

++++++++=

∑

∑

∑

∞

=

∞

=

∞

=

L

L

L

1 1 1 11

8

1

8

1

4

1

4

1

4

1

4

1

2

1

2

11

1

91

81

71

61

51

41

31

2111

1

1

1

x

x

x

x

x

x REPASSAR

33

Expected waiting times

• Geometric

• Pareto

• gausian

34

Expected value of a Binomial

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Small Mathematical Expectation