1 Elementary probability

Transcription

1 1 Elementary probability Problem 1.1 (*) A coin is thrown several times. Find the probability, that at the n-th experiment: (a) Head appears for the first time (b) Head and Tail have appeared equal number of times (c) Head has appeared exactly two times (d) Head has appeared at least two times. Problem 1.2 (*) Four letters are placed randomly in four envelopes. Find the probability to find at least one letter in the correct envelope Problem 1.3 (*) Two correctors read a book. Corrector A finds 200 missprints, corrector B finds only 150 missprints. Of the found missprints 100 overlap. Estimate, how many missprints remain uncovered. Problem 1.4 (**) Witnesses at a court say truth with probability 1/3. (a) Witness A says that witness B lies. Find the conditional probability, that B says the truth. (b) A says that B denies that C lies. Find the probability that C says the truth. Problem 1.5 (*) There are three types of spiral galaxies: Sa (23%), Sb (31%) and Sc (46%). The probability to observe an explosion of a supernova in a galaxy is for Sa, for Sb and for Sc. (a) Find the probability to to observe the explosion in a galaxy of unknown type. (b) An explosion is observed in a galaxy. Find the a posteriori probability that the galaxy belongs to Sa, Sb or Sc.

2 2 Random variables I Problem 2.1 (*) A Gaussian signal x is rectified: Find the probability density of y. y = { x if x 0 0 if x < 0 Problem 2.2 (*) A random variable X is uniformly distributed over the interval [0, 1]. Find the probability densities of X 2, X, sin 2 ( πx 2 ). Problem 2.3 (*) Find the probability density for the characteristic function G(k) = cos ak. Problem 2.4 (*) Represent the first moments M 1,...,M 4 via the cumulants. Problem 2.5 (*) How the moments and cumulants are transformed under a linear transformation of the random variable y = ax+b? Problem 2.6 (**) One random variable is a function of another one y = f(x). Represent the average y via the moments of x. Problem 2.7 (***) Can one decide from the characteristic function, whether the probability distribution is absolutely continuous, discrete, or fractal?

3 3 Random variables II Problem 3.1 (*) N points are randomly scattered on the interval [0, 1]. Find the probabilities of the following events: 1) the maximal point is smaller than t (where 0 < t < 1) 2) the minimal point is smaller than s 3) all points lie between s and t Problem 3.2 (**) Independent identically distributed random variables x i have probability density w(x). Find the probability density of y = max(x 1,...,x n ). Hint: Find first the probability distribution of y Problem 3.3 (**) Deminstrate that 1 ρ xy 1 and in the case ρ xy = ±1 there exists a linear relation between x and y. Hint: Consider the variance of x ±y, where x = (x x )/ D x is the normalized random variable. Problem 3.4 (**) x i have gaussian distribution with average 0. Demonstrate x 1 x 2 x 3 x 4 = x 1 x 2 x 3 x 4 + x 1 x 3 x 2 x 4 + x 1 x 4 x 2 x 3 Hint: Consider an expansion of the characteristic function G(k 1,k 2,k 3,k 4 ) and determine the term k 1 k 2 k 3 k 4. Problem 3.5 (**) Prove that for Gaussian random variables with vanishing average the following relation(furutsu- Novikov-Formula) holds x k f( x) = x k x j f x j j Hint: use partial integration Problem 3.6 (**) For i.i.d. x i we define y = x x r where r is an integer random variable with distribution p r. Find the characteristic function of y.

4 4 Central limit theorem, random processes Problem 4.1 (**) Consider a random variable x that is defined modulo 2π (e.g. a phase or an angle). 1) Are the moments suitable for characterization of x? 2) How would you write the chatacteristic function of x? 3) Are cumulants of x well defined? 4) What would you suggest as an alternative to moments and cumulants? 5) Calculate the distribution of the sum of two independent variables defined modulo 2π: y = x 1 +x 2 mod 2π 6) Let y = x 1 +x x n mod 2π. What can you say on the distribution of y for large n (in analogy to the central limit theorem)? Problem 4.2 (*) x(t) is a stationary random process. Find the statistical properties (average, variance, correlation function, power spectrum) of the process y(t) = x(t+t) x(t). Find the cross-correlation y(t)x(t ). Problem 4.3 (*) How the autocorrelation function and the power spectrum are transformed under the following transformations of the random process (a) y = ax+b (b) y(t) = x(t/a) (c) y(t) = x(t ), where = constant (d) y(t) = x(ξ t), where ξ is a time-independent random variable. Is process y(t) ergodic? Problem 4.4 (**) Find the autocorrelation function of the process x(t) = acos(ωt+φ) where a, ω, φ are independent random variables and φ is uniformly distributed on (0,2π).

5 5 Random processes: Correlation properties Problem 5.1 (**) Develop a correlation theory of processes with discrete time, i.e. x t, t =..., 2, 1,0,1,2,... 1) define the correlation function K t 2) define the power spectrum S(ω). To which range belong the frequencies ω? 3) Derive the Wiener-Khinchin theorem. Remains the relation variance = integral ofer the power spectrum valid? Problem 5.2 (*) The relation between the output signal y(t) and the input signal x(t) is given as (a) ẏ + y τ = ẋ (b) y(t) = 1 2T t+t t T x(u) du Find the relation between power spectra S y (ω) and S x (ω). Find the autocorrelation of y(t) in the case x(t) is white noise. Problem 5.3 (*) x(t) is a stationary Gaussian process with zero average. Find the autocorrelation function and the power spectrum of the process y(t) = x 2. Find the power spectrum of y in the case K x (t) = e t γ cosω 0 t. Problem 5.4 (**) Prediction a) One whants to predict the value of the stationary process x(t) (whith x = 0) at time t+t given the value at time t, with the help of a linear prediction ˆx(t+T) = ax(t). Find parameter a by minimizing the error Err = (ˆx(t+T) x(t+t)) 2. (This minimization is used also in all tasks below). b) Now also the derivative of the process at time t is used for the prediction: ˆx(t + T) = ax(t)+bẋ(t). Find a and b. c) One observes a process y(t), that is correlated with with x(t), and whants to predict the value of x using y: ˆx(t+T) = ay(t). Find a. d) One whants to estimate x(t) provided all the process y(t), < t < is known, using ˆx(t) = h(τ)y(t τ)dτ Fin the function h(t). Hint: Consider first the case of discrete time ˆx k = l a ly k l and demonstrate, that the optimal a l obeys (x k a l y k l )y m l Then write the corresponding equation for the time-continuous case.

6 6 Diffusion, Pulse processes Problem 6.1 (**) A complex variable z is governed by equation dz dt = iωz +x(t) where x(t) is a stationary random process. Find the diffusion constant of z: z(t) 2 Dt. Problem 6.2 (**) The process x(t) is governed by equation dx dt = a(x(t T) x(t))+ξ(t) where ξ(t) is a stationary random process. Find the diffusion constant of x. Problem 6.3 (**) (a) Process x(t) is a series of δ-pulses appearing at random times x(t) = δ(t t i ) Write a general expression for the power spectrum of x in the case when the time intervals between the pulses τ i = t i+1 t i are i.i.d. random variables with density f(τ). Demonstrate, that when all time intervals are close to τ 0 (e.g. if f(τ) is a Gaussian distribution with mean τ 0 and standard deviation τ << τ 0 ) the spectrum possesses sharp peaks. (b) Find the power spectrum of process y(t) = ( 1) i δ(t t i ) where the time intervals are defined as above. (c) Find the power spectrum for the generalized telegraph process t z(t) = y(t ) dt

7 7 Markov processes, Master equation Problem 7.1 (*) A generalized telegraph process has different rates α and β for transitions 1 1 and 1 1. Formulate the master equation and sove it. Find the average and the autocorrelation function of the process. Problem 7.2 (**) Prove that the stationary Gaussian Markov process is the Ornstein-Uhlenbeck process. 1) First express the transition probability density for a stationary Markov process via the correlation function 2) Insert this expression in the Chapman-Kolmogorov equation and derive the following equation for the correlation function: K(t+t ) = K(t)K(t ) 3) Write a differential equation for K(t) (Hint: consider the limit t 0) and solve it. Problem 7.3 (*) Consider a chemical reaction (death process) A k B Let x be the number of molecules of type A (k is the reaction rate). 1) Show that the master equation reads P(x,t) = kxp(x,t)+k(x+1)p(x+1,t), x = 0,1,...x 0 ; P(x,o) = δ x,x0 t 2) Show that the generating function F(s,t) = P(x,t)s x obeys F t = k(1 s) F s 3) Use the Ansatz F = f(g(s)λ(t)) and demonstrate that a general solution reads F(s,t) = f( ekt 1 s ) Show that the solution mit the given initial conditions is 4) Show that the probability reads 5) Demonstrate that F(s,t) = (1 e kt +se kt ) x 0 P(x,t) = x = x 0 e kt, ( ) x0 e x0kt (1 e kt ) x 0 x x (x x ) 2 = x 0 e kt (1 e kt ) At which value of x is the variance maximal? Show that for x 0 the process is like a deterministic one.

8 8 Fokker-Planck-Equation Problem 8.1 (*) Consider a particle in a bistable potential U(x) (e.g. U(x) = x 2 +x 4 ) Find the following transition probabilities ẋ = U x +ξ(t) ξ(t)ξ(t ) = 2σ 2 δ(t t ) lim limp(x,t x 0,0) t σ 0 lim lim P(x,t x 0,0) σ 0 t Problem 8.2 (**) The phase variable 0 φ < 2π evolves according to the Fokker-Planck equation P(φ,t φ 0,0) t = B 2 2 P(φ,t φ 0,0) φ 2 Find the solution of this equation in form of Fourier series in φ. Assuming stationarity, find the autocorrelation function of the observable x(t) = cos φ(t). Problem 8.3 (**) For the Langevin equation ẋ = x+ξ(t) find the probability to escape to + from an initial point x 0. Hint: Write the FPE with a constant source at x = x 0 Problem 8.4 (**) The price of a Ferrari car is M Euro, and the simplest way to collect money is to play a coin game. The starting capital is Q Euro, and then in the game there are probabilities α to win and β to loose one Euro at each step. The game goes on until either capital is Null (bankrupt) or M (enough for Ferrari). Find the probabilities of these events. (Use the result of (b) from previous problem). Hint: derive the Fokker-Planck-Equation and find probabilities to arrive at the corresponding levels.

9 9 First passage problem, ratchet Problem 9.1 (**) Intermittency of type I in presence of noise is described with the Langevin equation ẋ = ǫ+x 2 +ξ(t) ξ(t)ξ(t ) = 2σ 2 δ(t t ) A laminar stage begins at x = and ends at x =. Find anaverage duration of the laminar stage in dependence on ǫ, σ. Give also an asymptotic expression valid as σ 0. Problem 9.2 (***) ConsideraMarkovchainwithtransitionprobabilitymatrixp ik.calculatethemeanfirstpassage time T ij for a transition i j by summing the weighting the mean times of going directly i j and as i k j with the corresponding probabilities, and obtain an equation T ij = 1+ k j p ikt kj. Problem 9.3 (***) Develop a first-passage-time theory for the master equation with discrete states. Find an equation for the mean first passage time from state k to state n. Problem 9.4 (*) Calculate the flashing ratchet. At the noise-free stage the equation of motion reads ẋ = U x with potential U(x) = x/a for 0 x a and U(x) = x/b for b x 0; the spatial period of the potential is L = a+b. At the stage II there is noise but no potential: ẋ = ξ(t) ξ(t)ξ(t ) = 2σ 2 δ(t t ) Every stage is of duration T, where T a 2,b 2. 1) Find the transition probabilities x x±nl during the period 2T. 2) Find the averaged current and study its dependence on the asymmetry of the potential a b and on the noise intensity.