Tutorial 4

Cover: C2.9 Independent RVC3.1 Introduction of Continuous RVC3.2 The Exponential DistributionC3.3 The Reliability and Failure RateAssignment 4

Conica, Cui Yuanyuan

2.9 Independent Random Variables

Definition

2.9 Independent Random Variables

Mutually Independent

Pairwise Independent

1 2 1 21 2 1 2, , ( , , ) ( ) ( ) ( )r rX X X r X X X rp x x x p x p x p x=L L L

, ( , ) ( ) ( ) ,i j i jX X i j X i X jp x x p x p x i r j r= Î Î

2.9 Independent Random Variables

Pairwise independent of a given set of random events does not imply that these events are mutual independence .

Example

Suppose a box contains 4 tickets labeled by 112 121 211 222Let us choose 1 ticket at random, and consider the random eventsA1={1 occurs at the first place}A2={1 occurs at the second place}A3={1 occurs at the third place}P(A1)=? P(A2)=? P(A3)=?P(A1A2)=? P(A1A3)=? P(A2A3)=?P(A1A2A3)=?

QUESTION: By definition, A1,A2, and A3 are mutually or pairwise independent?

2.9 Independent Random Variables

Z=X+Y Xi Xj …Xr are mutually independent

2.9 Problem 1

2.9 Independent Random Variables

Z=max{X,Y}Z=min{X,Y} X and Y are independent

2.9 Problem 5

-( ) ( ),XF x P X x x= £ ¥ < <¥

( )( ) XdF xf x

dx=

3.1 Introduction of Continuous RV

CDF

pdf

0 0

0 1

1 1

, ,

( ) , ,

, ,

x

F x x x

x

ìï <ïïïï= £ <íïïï ³ïïî

1 0 1

0

, ,( )

, ,

xf x

otherwise

ì £ <ïï= íïïî

3.1 Introduction of Continuous RV

ExampleF(x)

x1

1

f(x)

x1

1

3.1 Introduction of Continuous RV

f(x)

x1

1

F(x)

x1

1

(f1) 0 for all

(f2) 1

(F1) 0 1

(F2) is a monotone increasing function of

(F3) 0 and 1

(F4) 0

( ) .

( ) .

( ) , .

( ) .

lim ( ) lim ( ) .

( ) ( ) .

x xc

c

f x x

f x dx

F x x

F x x

F x F x

P X c f x dx

¥

- ¥

®- ¥ ®+¥

³

=

£ £ - ¥ < <¥

= =

= = =

ò

ò

3.1 Problem 2

3.2 The Exponential Distribution

1 0

0

,( )

xe xF x

otherwise

l-ì - £ <¥ïïï= íïïïî

0

0

,( )

xe xf x

otherwise

ll -ì >ïïï= íïïïî

X ~ EXP( ) CDF & pdf

1 0

0

,( )

xe xF x

otherwise

l-ì - £ <¥ïïï= íïïïî

3.2 The Exponential Distribution

Interarrival time;Service time;Lifetime of a component;Time required to repair a component.

X ~ EXP( ) Example

3.2 The Exponential Distribution

X ~ EXP( ) Memoryless Property

X – Lifetime of a component t – working time until nowY – remaining life time

The distribution of Y does not depend on t.

e.x. The time we must wait for a new baby is independent of how long we have already spent waiting for him/her

Y ~ EXP( )

3.2 Problem 1

3.3 The Reliability and Failure Rate

3.3 Problem 1

Thanks for coming!Questions?