random variables and process.pptx

7/28/2019 random variables and process.pptx

1/21

Random Variables And Process

Prof. B B Tiwari


2/21

Introduction

Signals may be deterministic or random.

If uncertainty exists then signals are random

signals.

They are not predictable neither are they

completely unpredictable.

The probability of being correct can bepredicted up to certain extent.


3/21

Probability

When the possible outcomes of an experiment isnot always same we deal it with probabilitytheory.

For ex: when an experiment is repeated N timesand the possible outcomes A occurs NAThe relative frequency of occurrence of A is

NA /N .

It can be written asP(A)=

.


4/21

Totally independent events A1 and A2 with

probabilities P(A1) and P(A2) are mutually

exclusive events

P(A1 or A2) = P(A1) + P(A2)

In general

P(A1 or A2or .or AL) = ()

=1 and we know that

=1 = 1


5/21

If there are two events A and B one of the

events may affect the other .In this case the

conditional probability

P(B|A) = P(A|B)*P(B)/P(A)

This result is know as Bayes theorem.

If A and B are totally independent thenP(B|A) = P(B) and

Joint probability P(A,B) = P(A)*P(B)


6/21

Cumulative Distribution Function

F(x) =

and probability density functionf(x)=

F(x)

PDF has the following properties

f(x)>= 0 for all x


7/21

Probability of outcome X being less then or equalto x1 is

P(X


8/21

P(x1


9/21

For two random variables X and Y the

probability that x


10/21

FXY(x,y) = P(X


11/21

A communication example

We want to transmit one of two possible

messages the message m0 that the bit 0 is

intended or the message m1 that the bit 1 is

intended.

When received , generates some voltage , say

r0 , which may be as simple as a dc voltage,

while m1 received generates a voltage r1.


12/21

P(r0|m0)=probability that r0 is received given

that m0 is sent,

P(r1

|m0

)=probability that r1

is received given

that m0 is sent,

P(r=|m1)=probability that r0 is received given

that m1 is sent,

P(r1|m1)=probability that r1 is received given

that m1 is sent,


13/21


14/21

Clearly if P(m0|r0)>p(m1|r0) then we shoulddecide that m0 is intended and if the inequality isreversed we should decide for m1.Altogetherthen our algorithm should be :

If r0 is received: Choose m0 if P(m0|r0)>P(m1|r0)

Choose m1 if P(m1|r0)>P(m0|r0)

If r1 is received: Choose m= if P(m0|r1)>P(m1|r1)

Choose m1 if P(m1|r1)>P(m0|r1)


15/21

A receiver which operates in accordance with

this algorithm is said to maximize the

posteriori probability (m.a.p) of a correct

decision and is called an optimum receiver .

P(r0|m0)P(m0)>P(r0|m1)P(m1)

P(r1|m1)P(m1)>P(r1|m0)P(m0)


16/21

Average value of random variable

The possible numerical value of the random

variable X are x1 ,x2x3,., with probabilities of

occurrence P(x1), P(x1), P(x1).

x1P(x1)N + x2P(x2)N+.=N () The mean or average value of all these

measurements and hence the average value of the

random variable is calculated by dividing the sumshown above by the number of measurements N.

X= E(X)=m= ()


17/21

Tchebycheffs Inequality

P(|X|>=)


18/21

Variance of a random variable

2 = E[(X-m)2] = 2

Writing (x-m)2 = x2 -2mx + m2 in the integral ofabove equation and integrating term by term ,

2 = E(X2) - 2m2 + m2

= E(X2) m2

The quantity itself is called the standard

deviation and is the root mean square (rms) valueof (X-m). If the average value m=0, then

2 = E(X2)


19/21

Since x2 >=0 and f(x) >= 0 for all x we have that

2 0

Can be written as2 >= 2 2

+

In the ranges -


20/21

Gaussian Probability Density

F(x)=1

/

X =

/

dx = m

E[(X-m)2] =

/

dx = 2


21/21

Error Function

Documents

random variables and process.pptx