important random variables
DESCRIPTION
Important Random Variables. EE570: Stochastic Processes Dr. Muqaibel Based on notes of Pillai See also http://www.math.uah.edu/stat http ://mathworld.wolfram.com Check ‘ pdf ’, ‘ cdf ’ commands in Matlab. Continuous-type random variables - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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.
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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
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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')
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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.
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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
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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
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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)
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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
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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
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'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
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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