civl 181tutorial 5 return period poisson process multiple random variables
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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
2
2 2
By spirit of
1 1 exp
22 1 1
Y ,X Y|X X
y Y X XY|X
Y Y
f y,x f y | x . f x
y / xf y | x
2
comparing to normal distribution:
1 1 exp
22
Zz
ZZ
zf z
2 2
is normal with
1
Y ,X
y Y X X
Y
f y,x
E Y | X x / x
Var Y | X x
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?