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10-401-03-RandomSignalsProcesses

# 10-401-03-RandomSignalsProcesses - Random Variables and...

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EE 401 Digital Communications Random Variables and Stochastic (Random) Processes Random variables Stochastic(Random) Processes, Gaussian process Noise in communication systems

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2 EE 401 Digital Communications Deterministic or random signals Deterministic signal no uncertainty on its value at any time explicit mathematical expressions sin(x), exp(x) Random signal some degree of uncertainty, => no explicit expression value of the signal cannot be known in advance But it could be described by some statistical parameters; Mean Variance Average power Spectral distribution of power etc.
3 EE 401 Digital Communications Experiment, event and outcomes An experiment with k -th possible outcomes Set of all possible outcomes: sample space, S Single or multiple sample points in S: event Single sample point: elementary event Occurence of an event makes occurence of another event impossible: mutually exclusive events s 1 s k event S

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4 EE 401 Digital Communications Examples: Discrete or Continuous Experiment-1: A coin is tossed (discrete) Outcomes: heads (H) or tails(T) Sample space: S={H,T} Experiment-2: A die is tossed(discrete) Outcomes: 1,2,3,4,5,6 Sample space: S={12,3,4,5,6} Set of all real numbers betweem 0 to 100 (continuous) Sample space: S={x: x is a real number between 0 and 100}
5 EE 401 Digital Communications Set notation<=>space Sets A, B and C Union Intersection Mutually exclusive sets S A B A C B A B C C A B C

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6 EE 401 Digital Communications Probability (Pr) axioms Sample space might be discrete (tossing a die) or continuous (measument of voltage at the output of a noise source) Probability measure P is a function: assigning a non- negative number to an event A in the sample space S Three properties of probability 1.0 P [ A ] 1 2. P [ S ]= 1 3. A, and B are mutually exclusive events P [ A B ]= P [ A ] P [ B ] Tossing a coin three times Sample space? (TTT,..,HHH) Probability of two heads? P {2 heads} Probability of one head and two tails ? P {1 head and 2 tails}? s 1 s k S A B C 0 1 Pr events
7 EE 401 Digital Communications .. more on Pr axioms Additional axioms derived: 1.P [ A ]= 1 P [ A ] where A is the complement of the event 2.When A and B are NOT mutually exclusive , P [ A B ]= P [ A ] P [ B ]− P [ A B ] P [ A B ] is the joint probability 3.When A 1, A 2, ... , A m are mutually exclusive and all possible outcomes of an experiment , P [ A 1 ] P [ A 2 ] .. P [ A m ]= 1

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8 EE 401 Digital Communications Conditional Pr : Bayes’ theorem Conditional probability: an experiment involving the events A and B P [ B A ] :probability of B given that A is occured or conditional probability of B given A P [ B A ]= P [ A B ] P [ B ] resulting P [ A B ]= P [ A B ] P [ B ] P [ A B ]= P [ B A ] P [ A ] P [ B A ]= P [ A B ] P [ B ]/ P [ A ] Bayes' theorem , P [ A B ]= P [ B A ] P [ A ] P [ B ]
Appendix B1&B2 EE 401 Digital Communications Example: conditional Pr (Bayes’ theorem) Consider an experiment involving transmission of signals: s i : transmitted classes z j : received signals where i = 1,... ,M ,

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