signal detection part 1( prob&random)

7/30/2019 Signal Detection Part 1( Prob&Random)

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Signal Detection & Estimation ( Part 1)Review of Probability & Random Process

This part is a review of Probaility concpts and Random variable including

Processes. This is to bring the students to the same level of discussion further on

Signal Detection & estimation Theory.

Probability

Probability theory is an application of measure theory, though the relationship

between the two is more symbiotic . One by-product of the separate legacies of the

two theories is that the two have different names for the same concepts. In the

language of probability, the scenario in which we plan to fit a probabilistic model is

called the experiment. The set of all possible outcomes to this experiment is calledthe sample space ( ), and subsets of the sample space are called events()

Axiom 1. Given an experiment, there exists a sample space, , representing the

totality of possible outcomes of the experiment and a collection,A, of subsets, A,of called events.

Axiom 2. To each event A inA, there can be assigned a nonnegative number P(A)such that.

P(A) > 0 , P() = 1

A () ()

If A and B are two arbitrary events in the sample space , then

( ) () () ( )

Joint and Marginal Probability

Consider the collection of events A1,...,An. The probability that all of these occur is

called the joint probability of the events Aj and can be calculated through P(A1,A2

An) and above probability of arbitrary events .

Marginal Probability. Suppose that the sample space can be partitioned into two

different families of disjoint sets, {Ai } and {Bj }, i.e.,

( i=1 , j =1)

( ) is then the joint probability of events Ai and Bj .


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Hence we can write the marginal probability of Bj ( formed by taking the

sum of all joint probability of Bj with Ai.

() ( ) summation runs from i = 1 to m.

Conditional Probability

A concept that will be of very use in estimation theory is conditional probability. A

conditional probability is the probability of the occurrence of an event subject to the

hypothesis that another event has occurred.

P( B|A ) =( )

()

Also we can express () ( ) ( )

The probability of A given B is related to the probability of B given A through BayesRule

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

Another important concept when dealing with collections of events is the idea of

statistical independence. Two events A and B are independent if the occurrence of

one event gives us no information about the occurrence of the other. This manifests

itself in the conditional probability formula as

P(A|B) = P(A).

Two events, A and B, are orthogonal if one event implies that the other event will

not occur, i.e., P(B|A) = 0


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Random variable

Given a probability space, (,A,P),a random variable X() : Rn is a real (vector)

valued point function which carries a sample point, , into a point y Rn in sucha way that every set, A , of the form

A = { : X() x, x Rn }

A real random variable X() is said to be discrete if there exists a finitely countable

set S ={xj } such that

P ( : X() = xj ) = 1

Two random variables X and Y are called independent if any event of the form

X() Ais independent of any event of the formY() Bwhere A,B are sets in

Rn. The joint probability distribution of two independent random variables is the

product of their marginal distribution functions:

F(x,y) = P (X x, Y y ) = P (X x ) P (Y y )

= F(x)F(y),

and the same is true for the joint density function of two absolutely continuous

random variables:

fXY(x, y) = 2 / xy F |X=x,Y=y

= /x/y F|X=x F|Y=y

= {/x F|X=x }{ /y F|Y=y }

= fX (x) fY ( y)

Important

Suppose that we have a random variable, Y , defined as the sum of two

independent random variables, X1 and X2: Y = X1 + X2. If X1 and X2 have density

functions, fX1 and fX2 , respectively, then what is the density function of Y ?

Define Z1 = Y = X1 + X2 and Z2 = X 2 ( a linear relation with given vectors)

Z = AX now the derived Probability density can be written as

Z = gX = AX => X = g-1 (Z) = A-1 ( Z)

X = g -1 ( Z) ={

+ * + = { Z1 Z2 , Z2 }

T this finally terms out to

be the convolution integral.


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signal detection part 1( prob&random)

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