Class Material 04.pdf

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Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering [email protected]

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457.646 Topics in Structural Reliability In-Class Material: Class 04

ΙΙ

-5. Multiple Random Variables

 “Joint” Probability Functions e.g.

P X ( ≤ 20)

=

∫ dx

=

P ( ∩ ) = ?

Need more information than ( ) and ( )

① Joint Cumulative Distribution Function (CDF) (Discrete/Continuous) ↔ cf. __________ CDF

( , ) ( )

F

XY

x y ≡ P

•

F

_XY

( −∞ −∞ = , )

•

F

_XY

( , ) ∞ ∞ =

•

F

_XY

( −∞ , ) y

•

( , ) ( ) ( )

F

XY

∞ y = P ∩ = P

=

② Joint Probability Mass Function (discrete r.v’s) ↔ cf. __________ PMF .

(a) Definition :

P

_XY

( , ) x y ≡ P ( , )

(b)

F

_XY

( , ) a b = ∑

(c) Conditional PMF

( )

P

X Y

x y ≡ = =

(d)

P

_XY

( , ) x y → P x

_X

( ), ( ) ? P y

_Y

Question: Which one more likely?

Case A: Heavy & Tall

Case B: Light & Tall

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2 ( )

P x

X

=

∑

⇒( ) rule

(e) If X & Y are statically independent,

( )

P

X Y

x y =

( )

_Y

( ) P

Y X

y x P y

⇔

( , ) P

XY

x y

⇔

* In-class material on Joint PMF

③ Joint PDF (continuous r.v’s)

, 0

( , ) lim

XY x y

f x y

∆ ∆ →

=

(a) Joint cumulative distribution function (CDF)

( , ) ( , )

F

XY

x y ≡ P X ≤ x Y ≤ y

= ∫

( , ) f

XY

x y =

(b)

P a ( < X ≤ b c , < ≤ Y d ) =

(c) Conditional PDF

( )

f

X Y

x y

0

( )

lim

x

P x X x x

∆ →

x

< ≤ + ∆

= ∆

Can show

=

※ Multiplication rule

f

_XY

( , ) x y =

(s.i

f

_XY

( , ) x y =

)

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3

(d) Joint PDF→marginal PDF?

( )

f

X

x =

=

∫