11

Stochastic Structural Dynamics

Lecture-2

Dr C S ManoharDepartment of Civil Engineering

Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India

manohar@civil.iisc.ernet.in

Scalar random variables-1

Recall

• Uncertainty modeling using theories of probability and random processes

• Definitions of probability– Classical definition P(A)=m/n

– Relative frequency definition

– Axiomatic definition2

nmAP

n lim)(

3

ExperimentTrialOutcome

Randomexperiment

SampleSpace

Event e

Event space

0 1

R

E

B

Undefinednotions

0

1

if

iP A

P

P A B P A P BA B

Axioms

P e

•Sample point: element of sample space•Events are subsets of sample space on which we assign probability•Axiomatic definition does not prescribe how to assign probability

Axiomatic definition of probability

Recall (continued)

4

Recall (continued)

•Conditional Probability

•Stochastic independence

•Total probability theorem•Bayes theorem

.0;BAP

occurred has B given thatA event ofy Probabilit|Definition

BPBP

BAP

BPAPBAPBA tindependen areB and A:Notation

5

ExperimentTrialOutcome

Randomexperiment

SampleSpace

Event e

Event space

0 1

R

R X

0)(: (2)

event,an is )(: ,every for (1)such that line real into

space sample fromfunction a is variableRandom

XPxXRx

Random variable E

B

Undefinednotions

0

1

if

iP A

P

P A B P A P BA B

Axioms

P e

66

iX

xωX

i 10 Define654321

tossing.die of experiment heConsider t:Example

of Meaning

X is a subset of and hence

an element of and hence an event on which we assign probabilities.

x

B

Observation

54325020

1005

432140

XXXX

. writeWe xXxX

defined. are etc., ,covariance deviation, standardmean, assuch qunatitieson which basis theForms (c)

ies.uncertaint oftion quantifica Enables (b)y valued.numericallnot is which with deal tous Enables (a)

variable random a of notion the gintroducinfor Need

8

Definition

, ;

Notation, (the dependence on is not explicitly displayed)

X

X

X X

P x

P x P X x x

P x P x

Probability Distribution Function ︵ PDF ︶

9

2 1 2 1

2 1

2 1 1 2

1 1 2

2 1 1 2

2 1

( ) 0 ( ) 1( ) ( ) ( ) 1( ) ( ) ( ) 0( ) ( ) ( )PDF is monotone nondecreasing.Let .

X

X

X

X X

a P xb P P X Pc P P X Pd x x P x P x

x xX x X x x X x

X x x X x

P X x P X x P x X x

P X x P X x

Properties

0 0

( ) lim

PDF is right continuous.

X Xe P x P x

1010

x Xx PXx0 0-100 0600 15 010 1 1/620 1,2 2/622 1,2 2/635 1,2,3 3/653 1,2,3,4,5 5/6

: Die tossing: 1 2 3 4 5 6 ; 10iX i Example

1111

.continuous becomes T of PDF the,N As.1,,2,1 , at itiesdiscontinu have wouldT of PDF the,NFor

.1,,2,1; if T takeWe

.,,2,1,0; with

segments timediscrete into interval thedivide usLet

.arrival of time thedenoteswhich variablerandom thebe Let

likely.equally be interval in theinstant any timeLet

time.arrival TLet platform. on the

train a of arrival of timeheConsider t

1

1211

21

21

21

Nnt

Nntttt

NnttNntt,tt

N,tt

T,ttΩ

,tt

n

nnn

nnn

Example

12

lycontinuous and jumpsboth with proceeds PDF :Mixedjumps.any without proceeds PDF :Continuous

jumps.gh only throu proceeds PDF :Discretevariables Random

PDF of aMixed RV

xPxP XX

00

lim

1313

Probability density function (pdf)

x

XX

XX

duupxP

dxxdPxp

)()(

)()(

Definition

)(

)(

)(1)()(

dxxXxPdxxp

duupbXaP

duupPXPP

X

b

aX

XX

Properties

xpX

14

15

(a) ; 0,1,2, is also knwon as probability mass function (PMF).(b) PDF is also known as cumulative probability distribution function (CDF).

kP X k p k Notes :

Heaveside’s step function

16

ax

axaxaxU

21

1 0

x

axU

0,0

1

Example 1: Box function

17

bxUaxUxf )(

x

xf

0,0 a b

1

Dirac’s delta function

18

xxUdxd

aaxUxUaF

axUxUFxfaxUxUFxf

aFa

aFa

aFa

)(

lim

limlim)(

01

0

01

01

0

0

0

00

Dirac’s delta function

19

axaxUdxd

afdxaxxf

dxax

axax

1

for 0

Example-2: Stair case function

20

bxaxaxxdxdf

bxUaxUaxUxUxf

xf

x 0,0

a b

Commonly encountered random variables

• Models for rare events• Models for sums• Models for products• Models for extremes

– Highest– Lowest

• Models for waiting times

21

Limit theorems

2222

.1)1()1()()(

:Check).1()1()()1()1()(

110

Let FS

failure. and success :outcomes only two has experiment random The

--X

X

X

dxxpxpdxxp

xpxpxpxUpxpUxP

-p)P(XP(F)p )P(XP(S)

variable random Bernoulli

)(xpX

Remarks:• p is the parameter of the Bernoulli random variable.•Discrete random variable•Finite sample space•Basic building block

2323

Repeated Bernoulli trials: .The random experiment here consists of repeated Bernoulli trials.Assumptions(a) The random experiment consists of independent trials.(b) Each t

N

N

Binomial random variable

rial results in only two outcomes (success/failure)(c) P(success) remains constant during all trials.Define =number of successes in trials; 0,1,2,3, , .X N X N

(1 ) ; 0,1,2, ,N k N kkP X k C p p k N

0

Binomial theorem:

!( )! !

nn n r n r

rr

Nk

p q C p q

NCN k k

Notes

2424

Consider a sequence of trials resulting in successes.Occurrence of successes implies the occurrence of ( - ) failures.

Probability of occurrence of one such sequence= (1 ) .

Number of such pos

k N k

N kk N k

p p

0

0

sible sequences= .These sequences are mutually exclusive.

(1 )

(1 )

(1 ) 1

(By virtue of binomial theorem. )Hence the name binomial random variable

Nk

N k N kk

mN k N k

kk

NN k N k

kk

C

P X k C p p

P X m C p p

P X N C p p

.

25

A binomial random variable is denoted by ( , ). and are the paramters of this random variable.

B N pN p

Remarks

finite is space Sample variablerandom Discrete

26

xpX

xPX

x

x

10,0.5B

2727

Random experiments: as in binomial random variable.=number of trials for the first success; 1,2, , .first success in N-th trial (success on the -th trial failures on the

N NP P N

Geometric random variable

1

1

1 1

-1 -1

1 1

2 3

first ( -1) trials)

(1 ) ; 1,2, , .

(1 ) (this must be =1).

: let p=0.6

(1- ) 0.4 0.6

0.6 1+ 0.4 + 0.4 0.4

The expression inside the bracket is a g

n

n

n n

n n

n n

N

P N n p p n

P N n p p

Ex

p p

=

eometric progression.Hence the name geometric random variable.

systems gengineerin of timeslife modelingin Useful

space sample infiniteCountably variablerandom Discrete

28

xPX

xpX

Geometric random variable with p=0.4

x

29

( 1)( 1)

Define Number of trials to the -th success.It can be shown that

(1 ) ; , 1, 2, ,

k

w w k kk k

W k

P W w C p p w k k k

Pascal or negative binomial distribution

3030

(a) We are looking for occurrence of an isolated phenomenonin a time/space continuum. (b) We cannot put an upper bound on the number of occurrences.(c) A

Models for rare events : Poisson random variable

ctual number of occurrences is relatively small.

: goals in football match (time continuum), defect in a yarn(1- d space continuum), typos in a manuscript (2 - d continuum), defect in a solid (3 - d

Examples

0 0

continuum).

exp ; 0,1,2,!

exp exp exp exp 1.! !

k

k k

k k

aP X k a kk

a aP X a a a ak k

Stress at a point exceeding elastic limit during the life time of a structure.

Check

31

x

x

xPX

xpX

Poisson random variable with a=5

•Discrete RV•Countably infinite

sample space•Useful in wide variety of contexts