OP01 Random Variables

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Lectured by Ha Hoang Kha, Ph.D.Ho Chi Minh City University of Technology

Email: [email protected]

Random variables


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Content

Random variables Probability Histogram or probability density function Cumulative function Mean Variance Moments Some representations of random variables

Bi-dimensional random variables Marginal distributions

Independence Correlations Gaussian expression of multiple random variables

Changing random variables

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Examples : Acoustic waves Music,speech,

...

Light waves Light source(star, )

...

Electric current given bya microphone

Current given bya spectrometer

Number series Physical measurements

Photography ...

Introduction

s igna l = every entity which contains some physicali n fo rma t ion

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SpeechBiomedicalSound and MusicVideo and ImageRadar

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Typical Signals


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Signa l p roc ess in g = procedure used to:

extract the information (filtering, detection,estimation, spectral analysis...)

Adapt the signal (modulation, sampling.) (to transmit it or save it)

pattern recognition

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Signal Processing


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Dimension al c lass i f ica t ion : Number of free variables.

Examples :

Electrical potential V(t) = Unidimensional signal

Statistic image black and white brightness B(x,y) = bi-dimensional signal Black and white film B(x,y,t) = tri-dimensional signal...

Phenom enolog ical Class i f ica t ion Random or deterministic evolution

Deterministic signal : temporal evolution can be predicted or modeled by an

appropriate mathematical mode

Random signal : the signal cannot be predicted statistical description

The signal theory is independent on the physic phenomenon and the types ofvariables.

Every signal has a random component ( external perturbation, )

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Classification of Signals


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Morph olog ica l c lass i f icat ion :

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Probability

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Probability

If two events A and B occurs,

P(B/A) is the conditional probability

If A and B are independent, P(A,B)=P(A).P(B)

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, lim

lim lim lim /

AB

N

df AB A AB A

N N N A A

n P A B

N

n n n n P B A P A

n N n N

, / / P A B P B A P A P A B P B


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Let us consider the random process : measure the

temperature in a room

Many measurements can be taken simultaneously usingdifferent sensors (same sensors, same environments) andgive different signals

t z 1

z2

z 3

t 1 t 2

z , t x

Signals obtained when

measuring temperature

using many sensors

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Random variable and random process


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The random process is represented as a functionEach signal x(t), for each sensor, is a random signal.

At an instant t, all values at this time define a random variable

z ,

t x

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Random variable and random process

t z 1

z 2

z 3

t 1 t 2

z ,

t x

Signals obtained whenmeasuring temperature

using many sensors


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N(m, t i ) = number of events: " x i = x + D x"

x

m D x (m+ 1) D x

N(m)

Precision of measurment

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,Prob m 1 lim

,Prob lim

mes

mes

ii N

mes

p

ik mi N

mes

N m t m x x x

N

N k t m x x p x

N

D D

D D

N mes = total number of measurments

Probability density function PDF)

The characteristics of a random process or a randomvariable can be interpreted from the histogram

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PDF properties

x=dx, the histogram becomes continuous. In thiscase we can write:

0Prob 1 x x

2

1

1 2Prob , x

i i x

x x x f x t dx mes

i

N i N t xn

t x f mes

,lim, where

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Histogram or PDF

Random signal Sine wave :

Uniform PDF 14

-1 1

f(x)

x

Uniform PDF


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Cumulative density function

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Examples

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,i i E g g x f x t dx

Statistical parameters :

Average value :

Mean quadratic value:

, x i i it E x x f x t dx

Variance : 22 22 x i i x i x i xt E x t m t

Standard deviation : 2 2

x i x xt m

Expectation, variance

Every function of a random variable is a random variable. If we know the

probability distribution of a RV, we can deduce the expectation value of

the function of a random variable:

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2

2 2 ,i i i x

m t E x x f x t dx


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Exponential random variable

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Examples


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Examples


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Examples


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Gaussian random variable

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( ) 2

x m

f x e

2( ) ( ) E X m V X

( ; ) (0;1) X m If X N m N


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Examples


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Bi-dimensional random variable

Two random variablesX and Y have acommon probabilitydensity functions as :

(X,Y) f XY (x,y) is the probability densityfunction of the couple

(X,Y)

2 2( )( , ) x y XY f x y ce

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Bi-dimensional Random variables

Cumulative functions:

Marginal cumulative distribution functions

Marginal probability density functions

dxdy y x f yY x X P y x F x y

Y X Y X

),(),(),( ,,

1),(, dxdy y x f Y X y x

y x F y x f Y X

),(),(

2

,

dxdy y x f y F y F

dxdy y x f x F x F

y

Y X XY Y

x

Y X XY X

),(),()(

),(),()(

,

,

duvu f v f

dvvu f u f

XY X

XY X

),()(

),()(

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Examples


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Bi-dimensional Random variables

Moments of a random variable X

,

, ,

[ ] ( , )

[ ] ( , ) ( , ) ( )

[ ] ( )

[ ] [ ]. [ ]

n n n n X Y

X Y X Y X

y

E x y x y f x y dxdy

E X xf x y dxdy x f x y dydx xf x dx

E Y yf y dy

E XY E X E Y

, ( , ) ( ). ( ) X Y X Y f x y f x f y

If X and Y are independents and in this case

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Covariance

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Covariance

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Correlation coefficient

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PDF of a transformed RV


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PDF of a transformed RV

Suppose X is a continuous RV with known PDF

Y=h(X) a function of the RV XWhat is the PDF of Y?

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PDF of a transformed RV: exercises


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PDF of a transformed RV: exercises

X is a uniform random variable between -2 and 2. Write the expression on pdf of X Find the PDF of Y=5X+9

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Sum of 2 RVs


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Sum of 2 RVs

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Sum of 2 RVs


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Sum of 2 RVs

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Sum of 2 RVs


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Sum of 2 RVs

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Sum of 2 RVs


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Sum of 2 RVs

OP01 Random Variables

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