Stochastic signals and processes

1. Lecture 1 Stochastic signals and processes welcome

2. Introduction to probability and random variables • A deterministic signal can be derived by mathematical expressions. • A deterministic model (or system) will always produce the same output from a given starting condition or initial state. • Stochastic (random) signals or processes • Counterpart to a deterministic process • Described in a probabilistic way • Given initial condition, many realizations of the process

3. Introduction to probability • Define the probability of an event A as: • where N is the number of possible outcomes of the random • experiment and NA is the number of outcomes favorable to the • event A. • For example: • A 6-sided die has 6 outcomes. • 3 of them are even, • Thus P(even) = 3/6

4. Axiomatic Definition of Probability • A probability law (measure or function) that assigns probabilities to events such that: • P(A) ≥ 0 • P(S) =1 • If A and B are disjoint events (mutually exclusive), i.e. A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B) • That is if A happens, B cannot occur.

5. Some Useful Properties • : probability of impossible event • , the complement of A • If A and B are two events, then • If the sample space consits of n mutually exclusive events such that , then

6. Joint and marginal probability • Joint probability: • is the likelihood of two events occurring together. Joint probability is the probability of event A occurring at the same time event B occurs. It is P(A ∩ B) or P(AB). • Marginal probability: • is the probability of one event, ignoring any information about the other event. Thus P(A) and P(B) are marginal probabilities of events A and B

7. Conditional probability • Let A and B be two events. The probability of event B given that event A has occured is called the conditional probability • If the occurance of B has no effect on A, we say A and B are indenpendent events. In this case • Combining both, we get , when A and B are indenpendent

8. Random variables • A random variable is a function, which maps events or outcomes (e.g., the possible results of rolling two dice: {1, 1}, {1, 2}, etc.) to real numbers (e.g., their sum). • A random variable can be thought of as a quantity whose value is not fixed, but which can take on different values. • A probability distribution is used to describe the probabilities of different values occurring.

9. Random variables • Notations: • Random variables with capital letters: X, Y, ..., Z • Real value of the random variable by lowercase letters (x, y, …, z) • Types: • Continuous random variables: maps outcomes to values of an uncountable set. the probability of any specific value is zero. • Discrete random variable: maps outcomes to values of a countable set. Each value has probability ≥ 0. P(xi) = P(X = xi) • Mixed random variables

10. Continuous random variables • Distribution function: • Properties: • 1. is either increasing or remains constant. • 2. • 3. • 4. • 5.

11. Distribution functions • Cumulative distribution function (CDF) for a normal distribution

12. Probability density function (pdf) • Definition: • Properties: • Thus integration of gives probability

13. Expectation operator • Is defined only for random variables or a function of them. • Expected value: Is the weighted average of all possible values that this random variable can take on. • Let

14. Definition of moments • A moment of order k of a random variable is defined as: • Order 1: • Order 2: Mean of X

15. Central moments • Defined in a similar way as moment Variance of X

16. Two random variables • Let X and Y be two random variables, we define the joint distribution function • and the joint probability density function as • ≥ 0

17. Useful Properties

18. Useful Properties • The probability that Xlies between x1 and x2 and Y lies between y1 and y2 is

19. Indenpendent and uncorrelated random variables • Let X and Y be two random variables, we say that • X and Y are indenpendent if • X and Y are uncorrelated if • Indenpendence => uncorrelation

20. Expected value • If X and Y are indenpendent we get The correlation

21. Uncorrelated • If we say X and Y are orthogonal

22. Covariance Correlation coefficient

23. Random processes • A random process may be viewed as a collection of random variables, with time t as a parameter running through all real numbers.

24. Example • With Φ a random variable uniformly distributed between 0 and 2π.

25. First and second order statistics • Expected value • Autocorrelation function • Properties: Stationarity, Ergodicity, Power spectrum

26. Homework • Workshop: Get familiar with the following terms • Probability density function, independency Autocorrelation, Stationarity, Ergodicity, Power spectrum. • Exercises on the web (click here)