Important Random VariablesEE570: Stochastic Processes

Dr. MuqaibelBased on notes of Pillai

See alsohttp://www.math.uah.edu/stathttp://mathworld.wolfram.com

Check ‘pdf’, ‘cdf’ commands in Matlab

Continuous-type random variables

1. Normal (Gaussian): X is said to be normal or Gaussian r.v, if

This is a bell shaped curve, symmetric around the parameter , and its distribution function is given by

where is often tabulated. Since depends on two parameters and , the notation will be used to represent the above CDF

.2

1)(22 2/)(

2

x

X exf

,2

1)(22 2/)(

2

x y

XxGdyexF

dyexG yx 2/2

21)(

)(xf X

x

Thermal noise : Electronics, Communications Theory

Normal (Gaussian)

• is referred to as Standard Normal r.v.• Under very general conditions the limiting

distribution of the average of any number of independent, identically distributed random variables is norma.

2. Uniform: if, ),,( babaUX

otherwise. 0,

, ,1)( bxa

abxf X

)(xf X

xa b

ab 1

QuantizationCoding Theory

𝐹 𝑋 (𝑥 )={ 0 𝑥<𝑎(𝑥−𝑎) /(𝑏−𝑎) 𝑎≤𝑥<𝑏

1 𝑏≤𝑥

( )XF x

xa b

1

Exponential

• If occurrence of events over non-overlapping intervals are independent, such as arrival time of telephone calls or bus arrival times at a bus stop, then the waiting time is exponential.• Memoryless Property of exponential distribution• Memoryless property simplifies many

calculations and is mainly the reason for wide applicability of the exponential model.

otherwise. 0,

,0 ,1)(

/ xexfx

X

Queuing Theory

-2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

x

f X(x)

prameter=1parameter=2

% Dr. Ali Muqaibelclose allclear allclc

x=-1:0.01:10y1=pdf('exp',x,1);y2=pdf('exp',x,2);plot(x,y1,x,y2,':');legend ('prameter=1','parameter=2')xlabel('x')ylabel('f_X(x)')Label ('Exponential Distribution')

4. Gamma: with (if

If an integer then

5. Beta: if

where the Beta function is defined as

otherwise. 0,

,0 ,)()(

/1

xexxf

x

X

)!.1()( nn

),( baX )0 ,0( ba

otherwise. 0,

,10 ,)1(),(

1)(

11 xxxbaxf

ba

X

),( ba

1

0

11 .)1(),( duuuba ba

x

)(xf X

x10

)(xf X

Queuing TheoryGamma is a generalization of the exponential distribution with two parameters . If , we get the exponential r.v.

6. Chi-Square: if (Fig. 3.12)

Note that is the same as Gamma

7. Rayleigh: if (Fig. 3.13)

8. Nakagami – m distribution:

),( 2 nX

)(2 n ).2 ,2/(n

otherwise. 0,

,0 ,)(22 2/

2 xexxf

x

X

(3-36)

(3-37)

,)( 2RX

x

)(xf X

Fig. 3.12

)(xf X

xFig. 3.13

otherwise. 0,

,0 ,)2/(2

1)(

2/12/2/ xex

nxfxn

nX

22 1 /2 , 0( ) ( )

0 otherwiseX

mm mxm x e x

f x m

Wireless Communications

In communication systems, the signal amplitude values of a randomly received

signal usually can be modeled as a Rayleigh distribution

9. Cauchy: if (Fig. 3.14)

10. Laplace: (Fig. 3.15)

11. Student’s t-distribution with n degrees of freedom

. ,1)2/(

2/)1()(2/)1(2

tnt

nnntf

n

T

,),( CX

. ,)(

/)( 22

xx

xf X

. ,21)( /|| xexf x

X

)(xf X

x

Fig. 3.14

x

)(xf X

Fig. 3.15t

( )Tf t

Fig. 3.16

Related to Gaussian, Comm. Theory

12. Fisher’s F-distribution/ 2 / 2 / 2 1

( ) / 2

{( ) / 2} , 0( ) ( / 2) ( / 2) ( )

0 otherwise

m n m

m nz

m n m n z zf z m n n mz

(3-42)

The exponential model works well for inter arrival times (while the Poisson distribution describes the total number of events in a given period)

Other distributions:Erlang (traffic), Weibull (failure rate), Poreto ( Economics , reliability), Maxwell (Statistical)

Discrete-type random variables

1. Bernoulli: X takes the values (0,1), and

2. Binomial: if (Fig. 3.17)

3. Poisson: if (Fig. 3.18)

.)1( ,)0( pXPqXP (3-43)

),,( pnBX

.,,2,1,0 ,)( nkqpkn

kXP knk

, )( PX

.,,2,1,0 ,!

)( kk

ekXPk

k

)( kXP

Fig. 3.1712 n

)( kXP

Fig. 3.18

The total number of favorable outcomes is binomial r.v.

The number of occurrence of a rare event in a large number of trials: e.g number of telephone calls at an exchange over a fixed duration

4. Hypergeometric:

5. Geometric: if

6. Negative Binomial: ~ if

7. Discrete-Uniform:

.,,2,1 ,1)( NkN

kXP

),,( prNBX1

( ) , , 1, .1

r k rkP X k p q k r r

r

.1 ,,,2,1,0 ,)( pqkpqkXP k

)( pgX

, max(0, ) min( , )( )

m N mk n k

Nn

m n N k m nP X k

The number of trials needed to the first success in repeated Bernoulli trials is geometric

The number of trials needed to the success in repeated Bernoulli trials is negative binomial

'beta' or 'Beta', 'bino' or 'Binomial', 'chi2' or 'Chisquare', 'exp' or 'Exponential', 'ev' or 'Extreme Value', 'f' or 'F', 'gam' or 'Gamma', 'gev' or 'Generalized Extreme Value', 'gp' or 'Generalized Pareto', 'geo' or 'Geometric', 'hyge' or 'Hypergeometric', 'logn' or 'Lognormal', 'nbin' or 'Negative Binomial', 'ncf' or 'Noncentral F', 'nct' or 'Noncentral t', 'ncx2' or 'Noncentral Chi-square', 'norm' or 'Normal', 'poiss' or 'Poisson', 'rayl' or 'Rayleigh', 't' or 'T', 'unif' or 'Uniform', 'unid' or 'Discrete Uniform', 'wbl' or 'Weibull'.

MatlabCheck ‘pdf’, ‘cdf’ commands in Matlab

Check rand, randn

Converting Data to PDF% Dr. Ali Muqaibelclose allclear allclcn=10000;f=randn(1,n);[y,x]=hist(f,10);y=y/n/(x(2)-x(1)); xm=-1:0.01:10;ym=pdf('norm',xm,0,1); [yr,xr]=ksdensity(f); plot(x,y,xm,ym,xr,yr,':'); legend ('Hist','Model','KSDensity')xlabel('x')

-5 0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

HistModelKSDensity

Matlab live demo• Impact of number of points• Difference between histogram

and pdf ; Normalization • Fitting