civl 181tutorial 5 return period poisson process multiple random variables

CIVL 181 Tutorial 5

Return period

Poisson process

Multiple random variables

If P (exceedence within the life time of the building, i.e., 10 years) = 0.1

Return period T = 100 years?

A question on return period

1. The r.v. is continuous or discrete?

2. What is the relation between Poisson and binomial?

3. v / vt?

Poisson process

1x n xn t t

x n n

limn

( )

!

xtt

ex

Bernoulli Sequence

Poisson Process

Interval Discrete Continuous

No. of occurrence Binomial Poisson

Time to next occurrence Geometric Exponential

Time to kth occurrence Negative binomial Gamma

Comparison of two families of occurrence models

Joint and marginal PDF of continuous R.V.s

Surface = fX,Y (x,y)

fX,Y (x=a, y)

fX,Y (x, y=b)

fY (b) = Area

fX (a) = Area

marginal PDF fX (x)

marginal PDF fY (y)

y =b

x=a

Joint PDF

Conditional PDF of Y given x=a fY|X(y|x =a)

a) Calculate probability

,

( , )

( , )d b

X Yc a

P a X b c Y d

f x y dxdy

,

( )

( , )X Ya

P a X

f x y dxdy

b) Derive marginal distribution

,( ) ( , )X X Yf x f x y dy

,( ) ( , )Y X Yf y f x y dx

c) Conditional distribution

,|

( , )( | )

( )X Y

X YY

f x yf x y

f y

Example: Bivariate normal distribution (3.55)

A formal def of bivariate normal distribution is:

also by arithmetic we can rewrite as

Find P(4 <Y< 6) if fX(x) is N (3,1), fY(y) is N (4,2)

= 0.2 when x = 3, 3.5, 4

2 2

22

1 -1 exp 2

2(1- )2 1

- < x,y < ; corrleation coeff

X X Y YX ,Y

X X Y YX Y

x x x xf x, y

22

2 2

1 1 1 1 exp exp

2 22 2 1 1

y Y X XXX ,Y

XX Y Y

y / xxf x, y

Compare to

(Double integral!)

2

6

2 24

3 5

3 5 4 6

1 1 exp

22 1 1

y Y X X

Y YX .

P X . , Y

y / xdy

Take x = 3.5 as example

,|

( , )( | )

( )X Y

X YY

f x yf x y

f y

2 2

2

And

3: 4 0 2(2/1)(3-3) = 4

3 5: 4 0 2(2/1)(3.5-3) = 4.2

4: 4 0 2(2/1)(4-3) = 4.4

1

2 1 0 04 3 84

s.d. (Y) = Var(Y

y Y X X

Y

E Y | X x / x

x .

x . .

x .

Var Y | X x

. .

) 3 84 1 95. .

(Take x = 3.5 as example)

(Take x = 3.5 as example,)

Knowing fY|X(y|x) = N (4.2, 1.95)

P(X = 3.5, 4 <Y< 6) = 0.361

Try X = 4, X = 3 as exercise

26

2 24

1 1 Now exp

22 1 1

6 4 2 4 4 2

1 95 1 95

0 92 0 10

0 361

y Y X X

Y Y

y / xdy

- . - .. .

. .

.

1.95

4.2

Ex 3.58The daily water levels (normalized to respective full condition) of 2 reservoirs A and B are denoted by two r.v. X and Y have the following joint PDF:

26 5

0 1

X ,Yf x, y / x y

x, y

(a) Determine the marginal density function of daily water level for reservoir A

(b) If reservoir A is half full on a given day, what is the chance that water level will be more than half full?

(c) Is there any statistical correlation between the water levels in the two reservoirs?

civl 181tutorial 5 return period poisson process multiple random variables

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