Chapter 3 Foundation of Mathematical Analysis § 3.1 Statistics and Probability § 3.2 Random Variables and Magnitude Distribution § 3.3 Probability Density Functions and Usages § 3.4 Stochastic Processes and Representation R. J. Chang Department of Mechanical Engineering NCKU

§ 3.1 Statistics and Probability(1) 1.Introduction (1) Terminologies (a) Statistics Randomization, Outcome, Events (b) Probability Axioms, Probability space, Random processes (c) Experiments Sample space, Probability measure Ex: Die experiment, Coin experiment

§ 3.1 Statistics and Probability(2) (2)Mathematical Model

§ 3.1 Statistics and Probability(3) 2. Probability Space (1) Definition In probability theory, the probability space is defined as a triplet {Ω, £, P}. Ω is called the sure event which is a space of points ω i. £ is called the sets of events and are subsets of Ω. P is called a probability measure.

§ 3.1 Statistics and Probability(4) Ex: Coin experiment-Two consecutive tosses of fair coin Outcomes (“Points”): ω 1 (H, H); ω 2 (H, T); ω 3 (T, H); ω 4 (T, T).

§ 3.1 Statistics and Probability(5) Specific Events: A i : Subsets of points. A 1 : At least a tail was thrown ω 2, ω 3, ω 4. A 2 : Exactly a tail was thrown ω 2, ω 3. A 3 : Exactly two tail were thrown ω 4.

§ 3.1 Statistics and Probability(6) (2) Probability measure Define P as a function mapping P: £  R, and P satisfy the following axioms. (a) P(A) ≧ 0, where A is an event and P(A) is called the probability of the event. (b) P(Ω)=1. (c) P(A ∪ B)=P(A)+P(B) provided that A,B £ and A∩B= Φ,Φ is called the impossible event.

§ 3.1 Statistics and Probability(7) (3) Structure of Probability Space

§ 3.2 Random Variables and Magnitude Distribution (1) 1. Random Variables and Distribution Function (1) Random variables: In a probability space (Ω ， £ ， P ), is a random variable if and only if X is measurable w.r.t the field £. (2) Distribution function: The function is defined as the distribution function of X (ω).

§ 3.2 Random Variables and Magnitude Distribution (2) (a) Fundamental properties of F( x ) A non-decreasing function with the properties:

§ 3.2 Random Variables and Magnitude Distribution (3) (b) Function Mapping

§ 3.2 Random Variables and Magnitude Distribution (4) 1. Probability Density Function (1) Definition p(x) is called the probability density function of x (ω) if If F(x) is differentiable w.r.t. x then

§ 3.2 Random Variables and Magnitude Distribution (5) Ex: Die Experiment Outcome: Six faces of the Die Ω = { f 1, f 2, f 3, f 4, f 5, f 6 } Subsets(total number): 2 6 Events: “Even”, Outcomes are f 2, f 4, f 6 Probability=3/6=1/2 Random Variable: Define X ( f i )=10 * i Face1, X ( f 1 )=10 Face2, X ( f 2 )=20 Face6, X ( f 6 )=60 …

§ 3.2 Random Variables and Magnitude Distribution (6) Distribution Function F (100) = P{ } = 1 F (35) = P{ } = 1/2 Probability Density Function —Uniform discontinuous function

§ 3.3 Probability Density Functions and Usages (1) 1. Densities and Distributions

§ 3.3 Probability Density Functions and Usages (2) 2. Fundamental Usages

§ 3.3 Probability Density Functions and Usages (3) Ex: Gaussian density and probability distribution

§ 3.4 Stochastic Processes and Representation (1) 1. Time Domain Representation (1) Stochastic processes A stochastic process X(t, ω) is a family of random variables defined on the probability space {Ω ， £ ， P} and indexed by time t. For fixed time, we obtain a random variable which is measurable w.r.t. £. For each ω, we obtain a function mapping X: t  R called a sample function.

§ 3.4 Stochastic Processes and Representation (2)

§ 3.4 Stochastic Processes and Representation (3) (2) Gaussian process A random process is a Gaussian process if for any finite collection of n parametric values at t 1, t 2 … t n, the corresponding n random variables X(t 1 ), X(t 2 ) … X(t n ) are jointly Gaussian. The probability density function of the random variables X 1, X 2, X 3 … X n can be given by p(x 1, t 1 ; x 2, t 2 … x n, t n ):

§ 3.4 Stochastic Processes and Representation (4) (3) Properties of Gaussian process (a) Invariant property (b)Ergodicity property For weakly stationary (up to 2 nd moment) process

§ 3.4 Stochastic Processes and Representation (5) 2. Frequency Domain Representation (1) Magnitude representations (a) Amplitude spectrum (b) Energy spectrum (c) Power spectrum (d) Power spectral density

§ 3.4 Stochastic Processes and Representation (6) (2) Mathematical and physical spectrum Magnitude in dB:

§ 3.4 Stochastic Processes and Representation (7) 3. Gaussian white process (1) Definition A Gaussian process v(t) define on {Ω ， £ ， P} is a white process if its mean and covariance functions are given by (a) E[v(t)]=0 (b) E[v(t) ‧ v(s)]=Qδ(t-s) (2) Interpretation of whiteness Frequency – domain interpretation Power spectrum is Fourier transform of autocorrelation function.

§ 3.4 Stochastic Processes and Representation (8) (3) Role of Gaussian white process (a) Mathematics: a model of ideal random signal source (b) Physics: a model of physical noise (c) Engineering: signal for dynamic testing Local (Band limited) white

§ 3.4 Stochastic Processes and Representation (9) Gaussian white and colored process