ST2004 2011 Week 10 2

Systems• Teams in league– Games won by NA N B NC

– Sums of binary• Student attendance at class– Binary Name Chosen/ Not

Chosen– Given chosen Presence/Absence

• Sets of Dice– Scores X1 X 2 X3 Sums, Max S 3 M3

ModelDecompose

SplashBean Machine

ST2004 2011 Week 10 6

Systems of Random Variables

• Input Output– Simulation

• Joint Uncertainty (Input) Uncertainty(Output)

– Prob Dist

– Expected Values and VariancesIndependentDependentLinearNon-linear

ST2004 2011 Week 10 7

Linear Combs & Normal Distribution

• Linear Combinations – weighted sums– counts

• Simple for Normal• Normal a useful approx– Central Limit Theorem– SE(mean&prop) n Convergence

Dice sumDice max

ST2004 2011 Week 10 9

Prob Rules Prob Dist ExpVal etcMini League

Define = Number of wins by

twice as good as , ;

, evenly matched

Seek joint prob dist of ( , )

prob dist of

[ ], [ ]

prob dist of

joint prob dist of ( , )

[ , ]

B

A B

A B

A B

N B

A B C

B C

N N

S N N

E S Var S

D N N

S D

Cov S D

Joint NA Marg DistDist 0 1 2 NB

0 0 0.111 0.222 0.333NB 1 0.056 0.222 0.222 0.500

2 0.056 0.111 0 0.167NA 0.112 0.444 0.444 1

Cond Dists for NA0 1 2

Given 0 0.000 0.333 0.667 1.667 0.22NB= 1 0.112 0.444 0.444 1.332 0.45

2 0.335 0.665 0.000 0.665 0.22

Marg Dist

Cond Exp Vals

Cond Vars

ST2004 2011 Week 10 12

Max and Min CombinationsA system has 3 components A, B and C, with redundancy. It is designed such that it will work if either (C is working) or (both A and B are working). If the lifetimes of A, B and C are 10, 15 and 8 hours, resp, then it will work for 10 hours.

Components , ,

Sub-system , System

Random var identity

Event Id's

A B C

AB S

AB

S

AB

AB

S

S

T T T

T T

T

T

T t

T t

T t

T t

Probs

Times Exp Dist

Pr( )

Pr( )

Pr

Pr

Pr

Pr

~ ( )Comp

AB

AB

S

S

T

T t

T t

T t

T t

T t

T t

Exp mean

ST2004 2011 Week 10 15

Sum/Diff/Lin Comb indep random vars

• Expected Value & Var• simple rules, based on

– E[aX+bY] =aE[X] + bE[Y]– Var[aX+bY] =a2Var[X] + b2E[Y]

• cdf(system) pmf(system)– by tabulation, enumeration if discrete• some special cases

– intricate calculus, if continuous• some special cases

– but often, Normal approx

ST2004 2011 Week 10 17

Theory: Linear Combinations

X,Y random variablesa,b constantsZ = aX+bYSeek E[Z] and Var[Z]Using Normal (approx) for dist Z?

E[Z] and Var[Z] fully specifyDiscrete dists only in these notes; extension to continuous dists only a matter of notation; joint pdf instead of joint pmf; integrals instead of sums.

ST2004 2011 Week 10 18

Approach via dist Z =Y+X

31 2 127 27 27 3

32 2 127 27 27 3

3 2 1 127 27 27 331 2 2 1

9 9 9 9 9

Prob

and Poss 2 3 4 5 6

1 0 0

2 0 0

3 0 0

1

Y

sum

X

sum

E[Y] E[X]Var[Y] Var[X]E[Z]Var[Z]

InFill in given Indep

ST2004 2011 Week 10 19

Approach via dist Z =Y+X

3 6 7 6 31 127 27 27 27 27 27 27

Poss( ) are 3, 4, ...9

Prob( ) ?

Event Identities

( 3) ( 1, 2)

( 4) ( 1, 3) ( 2, 2)

( 3, 1)

Poss 3 4 5 6 7 8 9

Pr( )

Z z

Z z

Z X Y

Z X Y OR X Y

OR X Y

z

Z z

31 2 127 27 27 3

32 2 127 27 27 3

3 2 1 127 27 27 331 2 2 1

9 9 9 9 9

Prob

and Poss 2 3 4 5 6

1 0 0

2 0 0

3 0 0

1

Y

sum

X

sum

E[Y] = 2 E[X]=4Var[Y] =2/3 Var[X]=4/3E[Z]= 6 Var[Z]=6/3Cov(X,Y)=0

ST2004 2011 Week 10 23

Direct approach: when X,Y not indep

23

43

23

102 4 23 3 3 3

Without computing dist( )

[ ] 2 [ ]

4 [ ]

[ , ]

Use general result

[ ] [ ] 2 4 6

[ ] [ ] [ ] 2 ( , )

2

Z

E X Var X

E Y Var Y

Cov X Y

E Z E X Y

Var Z Var X Var Y Cov X Y

ST2004 2011 Week 10 24

2 2

2 2

2 2

[ ] [ ] [ ]

[ ] [ ] [ ] 2 [ , ]

[ , ] 0[ ] [ ]

[ ] [ ]

E aX bY aE X bE Y

Var aX bY a Var X b Var Y abCov X Y

a

in general

in general

whenCov X Y

when

Var X b Var Y

a Var inX b Var eY d p

Theory: Expected values for linear combs

ST2004 2011 Week 10 25

App: Travel Times

If time to travel A B,B C ,

Time to travel A C via B =

If told can model via:

~ 35,10 , ~ 45,25

Seek 1 Pr >10% longer than avg

2 Pr >10% longer than avg

3 Pr >10% longer than a

AB BC

AC AB BC

AB BC

AB

BC

AC

T T

T T T

T N T N indep

T

T

T

vg

80, 35, ~ 80,35AC AC AC

Theory

E T Var T T N

ST2004 2011 Week 10 27

App: Travel Times

Time to travel A C via B =

If told ~ 35,10 , ~ 45,25

Seek 1 Pr >10% longer than avg

2 Pr >10% longer than avg

3 Pr >10% longer than avg

80, 35, ~ 80,35

AC AB BC

AB BC

AB

BC

AC

AC AC AC

T T T

T N T N indep

T

T

T

Theory

E T Var T T N

ST2004 2011 Week 10 28

Travel Times

~ 35,10 ~ 45,25 ~ 80,35AB A ABT N T N T N

mean 35 45 80var 10 25 35Probs 0.134 0.184 0.088

2If ~ , then Z ~ 0,1

Event Identity

Y

y

Y N N

Y y Z

ST2004 2011 Week 10 29

Times Different?

~ 35,10 , ~ 45,25

Seek Pr >

- 10, - 35

0 ( 10) 10 - >0 Std Norm 1

35 35

Event Identity > - >0AB B

AB BC

AB BC

AB BC AB BC

AB BC

C AB BC

T N T N indep

T T

Theory

E T T Var T T

T T

T T T

Z

T

Pr = 0.045

ST2004 2011 Week 10 32

Proof: discrete case

,

, ,

[ ] (Poss( ) Probs)

= ( ) Pr( , )

= Pr( , ) Pr( , )

= Pr( , ) Pr( , )

= Pr( ) Pr( )

[ ] [ ]

x y

x y x y

x y y x

x y

E X Y Sum X Y

x y X x Y y

x X x Y y y X x Y y

x X x Y y y X x Y y

x X x y Y y

in gE X E e raY ne l

ST2004 2011 Week 10 34

Packing a pillbox

2

2

2

Mean Pill Depths 40 SD 3

Model Pill Depths ~ 40,3

Tube 420,5

Cork in Tube 15,2

Seek Pr(All fit)

Pill Tube designed to take 10 pills;

mm mm

N iid

N

N

ST2004 2011 Week 10 36

Extension

1 1 2 2 3 3 1 1 2 2 3 3

1 1 2 2 3 3

2

2

[ ...] [ ] [ ...]

[ ] [ ] [ ...]

General Result

i i i i

i i i i

i i i i

E a X a X a X a E X E a X a X

a E X a E X E a X

E a X a E X

Var

in general

whenindepa X a Var X

Va

In gene

r a X a Var X

ral

2 ,i j i j

i j

a a Cov X X

ST2004 2011 Week 10 38

Common Error

Let Pill Depth; Tube; Cork

(All fit) 10 0

10

420 (40 ... 40) 15 5

[ ] 25 100(9) 4 929

0 5( 0) Std Norm

929

X T C

T X C

Z T X C

E Z

Var Z

Z

ST2004 2011 Week 10 39

Important Special Cases

2 2

1 2

1 2

21 1

1 11 2

1 1

2 21 1 1

.....

[ ] .....

[ ] ..

.....

[ ]

[

increases at rate

always

decreases at ra] t[ ]

Total

Avg

n n

n n

n n

n n nn n

n nn n

n n nn n

Y X X X iid

E Y E X E X E X n

Var Y Var X Var X Var X n

X X X X Y

E X E Y n

Var X Var Y n

n

1

1

e

decreases at [ a e] r t

n

n nSD X

n

ST2004 2011 Week 10 41

Simulation Convergence

8.00

9.00

10.00

11.00

12.00

13.00

14.00

15.00

16.00

1 101 201 301 401 501 601 701 801 901

S4 lo

S4 hi

Running avg of S4

Confidence Intervals for Simulations Sec5.7

ST2004 2011 Week 10 43

Theory: Normal Approximationvia Central Limit Theorem

1

If indep with common [ ], [ ]

Then approx [ ], [ ]

1 1and approx [ ], [ ]

~

~

i

n

n i

n n

Y E Y Var Y

S Y N nE Y nVar Y

Y S N E Y Var Yn n

ST2004 2011 Week 10 44

Application: sums and averages

1 2

7 352 12

1

1

7 352 12

72

Let , .......... .. regular die

[ ] ; [ ]

Define

(scores on dice)= ; (scores on dice)

Recall [ ] ; [ ]

Thus [ ] ;

n

i i

n

n i n nn

i

n nn n

n

indep identically disX X X

E X Var X

Y sum n X X Y avg n

E Y V

tributed ii

a

E

d

r Y

X

3512[ ]n nVar Y

2

2

Normal Approx via Central Limit Theorem

7 35,

2 12

7 35,

2 12

~

~

n

n

n nY N

X Nn

ST2004 2011 Week 10 45

Application: precision

2

2

21 2

General Result

, .......... .. [ ] ; [ ]

Sample Mean (values of )

Show [ ] ; [ ]

Implications? used as estimator of

Precision of estimator 2

n i i

n i

n n n

n

n

X X X E X Var X

X avg X

E X Var Y

i

X

id