Random Variable - Stochastic Structural Dynamics - Lecture Slides, Slides of Stochastic Processes

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Scalar random variables-

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2

Recall

^

^

^

event,

an

is

every

for

such that

line

real

into

space

sample

from

function

a

is

variable

Random









X

P

x

X

R

x

•Discrete•Continuous•Mixed

Descriptions •Probability distribution function•Probability density function•Probability mass function

PDF of aMixed RV

 

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44

Models for rare events : Poisson random variable (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

ctual number of occurrences is relatively small.

: goals in football match (time continuum), defect in a yarn

Examples (1- d space continuum), typos in a manuscript (2 - d continuum),defect in a solid (3 - d

^

^

^



^

^

^

^

^

^

^

^

^



0

0

continuum).

exp

;^

exp

exp

exp

exp

!^

k

k^

k

k^

k

a

P X

k^

a^

k

k

a^

a

P X

a^

a^

a^

a

k^

k

^



^



^

^

^

^

^

^

^

^





Stress at a point exceeding elastic

limit during the life time of a structure.Check

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5

x x

  x

P

X

  x

p

X

Poisson random variable with a=

•Discrete RV•Countably infinitesample space•Useful in widevariety of contexts

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7

^

^

^

! 

(

)^

1

(^

)!!

!

(^

)!!

!

1

exp(

)

1

exp

exp(

) exp(

)^

exp(

)

(

)^

exp(

)^

exp(

)

!^

!

k^

n^

k

n^

k^

n^

k

k

k

n

k

k

k

n^

k

k

k^

k

n^

k^

n^

k

k

n^

a^

a

C p

p^

n^

k^

k^

n^

n

n^

n

C

n^

k^

k^

k

a

p^

n p^

p

p^

n^

k^

p^

a^

kp

a

n^

a^

a

C p

p^

a^

a

k^

n^

 k







^

 





^



^

 



^

^

 







 ^





^

 ^



^

^



^



^

^

^

^

^

^

^



^

 ^





^



^

^



^

 ^



Proof

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^



^

^

^

 

^

(^36769). 0

10 1

10

0

10

1000;

(^10) ,

1000

ExampleConsider

0 1000 3

0 3

0

1000

3

3





 





  





 



C

X P

p

N

B

^

^

^

^

^

^

(^36787). 0 1

exp ! 0

exp 0

1000

0

1

10

1000

0

3

 





   





^



a a

X Np k P

limit

Poisson

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1010

^  

x

m

x

x

p

X^

exp

(^

2

2



 

Gaussian (Normal) random variable: model for sums

and

are the paramters

of a Gaussian random variable.-^

;^

Gaussian random variable isdenoted by

(^

,^

)

m

m

N m 





 

 



  x p^ X

x

^

 8

. 0 , 2 N

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1111

=0 and

m



Standard normal random variable

^

 (^1) , 0 N

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^

^

^

(^500) 

Let

1000;

0.5;

1000

250;

15.81;

4

500

485

500

It is verified that 485

500

500

(

0.5)

1

1

(

exp

2

n 2 15.

p

k

npq

npq

np

npq

np

npq

k

P X

C





























 











^





^







^



^



^



^





^



^



^



Example

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1414

^

^

^



^

^

^



^

^

^



^

 ^



^

^

^

 

^





exp ;

Let

exp(

consider

Now,

in

points of

number :

Define

failure ;

success

interval- sub

in the lies point

success

Define

Let

is

randomly

points n place us Let

interval timea

Consider

2

1 2

1 2

2 1

1 2

2 1

2 1

k t k t

k X P

t a t t t

n T

a k a

p

p C

a

T

)t

n(t,

T

t t ,

n

p p C k X P

,tt

X

p

P

p

T

t t

P

,tt

T.

t t

,T.

,T

k a

a

a

a

k

kn

k k n

kn

k k n

times

waiting for

modelsl

exponentia

counting for

models

Poisson

points

Random

^

^ t^1

 t^2

0

T Docsity.com

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Transformation of random variables

If^

P,L,b,d,E

are RVs, what is the

pdf of

PL 3 EI

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1717

Let

X

be a RV; define

Y=g(X);

Given pdf of

X

, what is the pdf of

Y?

ies.

probabilit add to wish We

modulus? Why

^

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^



^

^

^

^

^



^

^

^

^

^

^

^



^

^

^

^

^

 y g x

dy dx

x p

y p

dy dx

x p

dy dx

x p

dy dx

x p y p

dx x p

dx x p

dx x p

dy y p

dy y Y y P

i

n i

i xx X

Y

xx X

xx X

xx X

Y

X

X

X

Y i

1

1

3

2

1

3 3 2 2 1 1

with

3

2

1













general In

Basic problem of transformation of random variables

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  x PX

  x p^ X

x x

LognormalRandom variable

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2020

^

^

y

ay

y a

p y a

p y p

ax

dy dx y a

aX x Y

X

X

Y^

2

Example

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