Lecture 1. Introduction to stochastic processes: concepts, definitions and applications.

Transcription

1 Lecture 1. Introduction to stochastic processes: concepts, definitions and applications. Jesper Rydén Department of Mathematics, Uppsala University Stationary Stochastic Processes Fall 2011

2 Example: Temperature in Uppsala Winter in Uppsala.

3 Example: Temperature in Uppsala Minimum daily temperature in Uppsala Temperature (C) Year

4 Example: Offshore technology The Draupner platform, North Sea.

5 Example: Offshore technology Hs (m) Measurements of significant wave height.

6 Example: Vehicle industry Lorry at test track.

7 Example: Vehicle industry Last Tid / s Loading of the rear axle of a lorry, loading and unloading gravel.

8 Example: Signal processing

9 Example: Signal processing

10 Definition, stochastic process Definition. A stochastic process with parameter space T is a family {X(t), t T} of random variables, defined on a sample space Ω. If T is a real interval, the process is said to have continuous time. If T is a sequence of integers, the process is said to have discrete time, and it is called a random sequence, or time series. Customary: suppression of the argument ω. A more complete notation for the function values would be X(t,ω). For a fixed outcome ω, the function t X(t,ω), is called a realization of the process. Other names: path, trajectory, sample function. The set of possible sample paths is called the ensemble of the process.

11 Random experiment

12 Random experiment: Roll a die Roll a die at times T = 0,1,2... and record the outcome N T, 1 N T 6. The random process X(t) is defined such that for T t < T +1, X(t) = N T.

13 Four sample functions of four kinds

14 Example: Production line In a production line for 1000Ω resistors, the actual resistance (in ohms) of each resistor is a random variable R, uniformly distributed on (950, 1050). The resistances of different resistors are independent. The resistor company has an order for 1% resistors with a resistance between 990Ω and 1010Ω. An automatic tester takes one resistor per second and measures its exact resistance (this test takes one second). The random process X(t) denotes the number of 1% resistors found in t seconds. The random variable T r (seconds) is the elapsed time at which r 1% resistors are found. (i) Find p, the probability that any single resistor is a 1% resistor. (ii) Find the probability-mass function of X(t). (iii) Calculate E[T 1 ], the expected time to find the first 1% resistor. (iv) Compute the probability that the first 1% resistor is found in exactly 5 seconds.

15 The Poisson process The Poisson process: A continuous-time process with discrete state space. A counting process, counting events from some starting time point: X(t) = Number of events in the time interval (0,t] where the variable X(t) increases by 1 each time an event occurs.

16 Definitions Definition. A stochastic process {X(t),t T} is said to have independent increments if X(t 2 ) X(t 1 ),X(t 3 ) X(t 2 ),...,X(t n ) X(t n 1 ) are independent for every choice of times t 1 t 2... t n in T have stationary increments if the distribution X(t +h) X(t) does not depend on t, only on h be simply increasing if it is non-decreasing with integer jumps and P(X(t +h) X(t) > 1)/h 0 as h 0

17 The Poisson process Definition. A simply increasing stochastic process {X(t), 0 t < } with X(0) = 0 and with stationary, independent increments is called a homogeneous Poisson process. One can prove that P(X(t +h) X(t) = k) = e λh(λh)k, k = 0,1,... k! where λ is called the intensity. Further, E[X(t)] = V[X(t)] = λt, [ ] X(t +h) X(t) E = λ. h

18 Example: A pipeline Consider an oil pipeline. Suppose the number of imperfections X(t) along a distance t can be modelled by a Poisson process, i.e. X(t) Po(λt) where λ is the intensity (km 1 ). One has found that λ = 1.7 km 1. Calculate the probability for more than two imperfections along a distance of 1 km.

19 Random variables, some properties and rules Theorem. Let a 1,...,a k and b 1,...,b l be real constants, and let X 1,...,X k and Y 1,...,Y l be random variables in the same experiment, i.e. defined on a common sample space. Then [ k ] k E a i X i = a i E[X i ], k C a i X i, i=1 i=1 [ k ] V a i X i i=1 = l b j Y j = j=1 i=1 k i=1 i=1 k i=1 j=1 k a i a j C[X i,y j ], l a i b j C[X i,y j ].

20 Definitions Let {X(t), t T} be a real-valued stochastic process with discrete or continuous time. For any stochastic process, the first and second-order moment functions are defined as m(t) = E[X(t)] mean-value function (mvf) v(t) = V[X(t)] variance function (vf) r(s, t) = C[X(s), X(t)] covariance function (cvf) b(s, t) = E[X(s)X(t)] second-moment function ρ(s, t) = ρ[x(s), X(t)] correlation function Some simple relations: r(t,t) = C[X(t),X(t)] = V[X(t)] = v(t), r(s,t) = b(s,t) m(s)m(t), C[X(s),X(t)] r(s,t) ρ(s,t) = = V[X(s)]V[X(t)] r(s,s)r(t,t)

21 Example Example. (a) Assume that X 1 and X 2 are random variables with V[X 1 ] = σ 2 1, V[X 2] = σ 2 2, C[X 1,X 2 ] = 3 and define Calculate V[Y]. Y = 2X 1 4X 2. (b) Assume now that X 1 and X 2 are independent random variables. Again, calculate V[Y].

22 Example Example. A stochastic process {X(t), t = 0,1,2,...} has the mvf m(t) = 8t and the cvf r(s,t) = 7min(s,t). Calculate E[X 1 +2X 2 ] and C(X 1 +2X 2,3X 3 +4X 4 ).

23 Example Example. Let X 1,X 2,X 3,... be iid variables with mean 6 and variance 17. Find the mvf and cvf for (a) the process {X t, t = 1,2,3,...} (b) the process {Y t, t = 1,2,3,...}, where (a summation process). Y t = X 1 +X 2 + +X t