1

Continuous random variables Continuous random variable

Let X be such a random variable Takes on values in the real space

(-infinity; +infinity)

(lower bound; upper bound)

Instead of using P(X=i) Use the probability density function

fX (t)

Or fX (t)dt

2

Cumulative function of continuous r.v.

The relationship between the Cumulative distribution of continuous r.v. and fX

=>

Properties for CDF

t

XX dssftF )()(

)()( tftFdt

dXX

3

Distribution function: properties

Properties for pdf

0][;0)(][)

][1)(][)

1)()

0)()

tXPdtdttfdttXtPd

tXPdttftXPc

dttfb

tfa

X

t

X

X

X

4

Uniform random variable

X is a uniform random variable

Mean:

Variance:

];[;0

;1

)(

];[

bat

btaabtf

baX

X

5

Exponential distribution

Exponential distribution is the foundation of most of the stochastic processes

Makes the Markov processes ticks

is used to describe the duration of sthg CPU service

Telephone call duration

Or anything you want to model as a service time

6

Exponential random variable

A continuous r.v. X Whose density function is given for

is said to be an exponential r.v. with parameter λ

Mean: and variance:

0;0

0;)(

][1;0

t

tetf

XE

t

X

7

Link between Poisson and Exponential

If the arrival process is Poisson # arrivals per time unit follows the Poisson distribution

With parameter λ

=> inter-arrival time is exponentially distributed With mean = 1/ λ = average inter-arrival time

Time

0 T

Exponentiallydistributed with 1/ λ

8

Proof

Number of arrivals in a t-second interval Follows the Poisson distribution with parameter

Let denote random time of first arrival

=> T is exponentially distributed

9

Memoryless Property

Proof:

10

Example

Suppose that amount of time you spend in bank is exponential with mean 10 min

What is the probability you spend more than 5 min in bank?

What is the probability you spend more than 15 min Given that you are still in bank after 10 min?

11

Hyper-exponential distributions H2 Hn

Advantage Allows a more sophisticated representation

Of a service time

While preserving the exponential distribution And have a good chance of analyzing the problem

λ1

λ2

p

1-p

.

.

λ1

λ2

λn

p1

p2

pn

12

Further properties of the exponential distribution

If are independent exponential r.v. With mean , then the pdf of is:

Gamma distribution with parameters n and

If and are independent exponential r.v. With mean and =>

𝑓 𝑋 1+𝑋 2+…+𝑋 𝑛(𝑡 )=𝜆𝑒−𝜆𝑡

(𝜆𝑡 )𝑛−1

(𝑛−1 )!

𝑃 (𝑋 1< 𝑋 2)=𝜆1

𝜆1+𝜆2

13

Further properties of the exponential distribution (ct’d)

X1 , X2 , …, Xn independent r.v. Xi follows an exponential distribution with

Parameter λi => fXi (t) = λi eλit

Define X = min{X1, X2, …, Xn} is also exponentially distributed

Proof fX(t) = ?

14

Joint distribution functions Discrete case

One variable (pmf) P(X=i)

Joint distribution P(X1=i1, X2=i2, …, Xn=in)

Continuous case One variable (pdf)

fX(t)

Joint distribution fX1, X2,…Xn, (t1,t2,.., tn)

15

Independent random variables The random variables X1 , X2

Are said to be independent if, for all a, b

Example Green die: X1

Red die: X2

X3 = X1 + X2

X3 and X1 are dependent or independent?

)().(),( bYPaXPbYaXP

16

Marginal distribution

Joint distribution Discrete case

P(X1=i, X2=j), for all i, j in S1xS2

=>

Continuous case fX1,X2(t1,t2), for all t1,t2

=>

2

),()( 211Sj

jXiXPiXP

221,1 ),()(211

dtttftf XXX

17

Expectation of a r.v.: the continuous case

X is a continuous r.v. Having a probability density function f(x)

The expected value of X is defined by

Define g(X) a function of r.v. X

dxxxfXE X )(][

dxxfxgXgE X )().()]([

18

Expectation of a r.v.: the continuous case (cont’d)

X1, X2, …, Xn: dependent or independent

Example:

)(...)()()...( 2121 nn XEXEXEXXXE

19

Variance, auto-correlation, & covariance Variance

Continuous case

If are independent r.v. =>

If X and Y are correlated r.v.:

Autocorrelation:

Covariance

20

Conditional probability and conditional expectation: d.r.v.

X and Y are discrete r.v. Conditional probability mass function

Of X given that Y=y

Conditional expectation of X given that Y=y

)(

),(

)(

),(

)|()|(|

yp

yxp

yYP

yYxXP

yYxXPyxp

Y

YX

x

yYxXPxyYXE )|(.]|[

21

Conditional probability and expectation: continuous r.v.

If X and Y have a joint pdf fX,Y(x,y) Then, the conditional probability density function

Of X given that Y=y

The conditional expectation Of X given that Y=y

)(

),()|(| yf

yxfyxf

YYX

dxyxfxyYXE YX )|(.]|[ |

22

Computing expectations by conditioning Denote

E[X|Y]: function of the r.v. Y Whose value at Y=y is E[X|Y=y]

E[X|Y]: is itself a random variable Property of conditional expectation

if Y is a discrete r.v.

if Y is continuous with density fY (y) =>

]]|[[][ YXEEXE

y

yYPyYXEYXEEXE ][].|[]]|[[][

dyyfyYXEXE Y )(]|[][

(1)

(2)

(3)

23

Proof of equation when X and Y are discrete

][][

],[],[

][][

],[

][].|[

][].|[]]|[[

XExXxP

yYxXPxyYxXxP

yYPyYP

yYxXPx

yYPyYxXxP

yYPyYXEYXEE

x

x yy x

y x

y x

y