Find the average p ower of the wsS random process X(t) which has the power spectral density o? -17 Sxx (@) (o +49)(@² +16)
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- Question 4 Let X be a random variable with AX) 0, EX) 0.6. if x 1 Answer (1) and (2): (1) If X, X2, X3, X4 are iid f, then (i) PX4) S0.8) = (.. ..). (ii) PX1) < 0) = (...). (2) Let X, X2,.., X36 be iid f You are given: PZS 1.96) = 0.975; P(ZS 2) = 0.977; P(Z< 1.55) 0.939, PZS 1.81) = 0.965. Use the CLT to approximate PXS 0.2). %3D %3D %3DProblem 1 Let X be a Poisson random variable with parameter 1 > 0, i.e. P(X = k) = xk k! Show that EX = A. Hint: Use that Σ/k! = e^. -k=0We assume that the car accident of a certain driver happen according to a Poisson N(t) of intensity lambda=0.01 per month. Compute the following (a) The probability that the driver has no accident during the first year (b) The probability of exactly one accident during his third year of driving. (c) the probability of at least two accidents before the end of the second year given that there was exactly one accident during the first year.
- Let a stochastic process (Xt) be given as follows: Xt = 3 + 0:5Xt-1 + t, where (t) is white noise with var ( t) = 1. a) What process is (Xt)? b) Explain briefly why this is a \conditional expectation" model, but not a \conditional variance" model. c) Suppose we observed Xt = 2:5, Xt-1 = 1:7. Compute a forecast for Xt+1. d) Suppose we observed Xt = 2:5, Xt-1 = 1:7. Compute a forecast for the variance of the process at time t + 1, that is, for the variance of Xt+1.In generating glossy rays, a and b are computed as do a = random In01 () b = random In01 () while a^2 + b^2 > 1 How many iterations, on average, does it take to find a and b? Provide an answer with three digits after the decimal. Note that, for a < 1, * 2 - 1 * 2 - 1 Σiai-1 i=1 A/ 1 (1 − a)²Example 9-53. (a) Find the p.d.f. of Xr) in a random sample of size n from the exponential distribution: f(x) = a e i, a>0, x 20. (b) Show that X) and W, = X9-X¢», r < s, are independently distributed. %3D
- Find auto-correlation function of a random process whose power spectral densis is given by 1+. 4Suppose X is a random variable whose pdf is given by f(x) =k(4x-2x^2) 01)EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({07 Find the Auto correlation function and power spectral densty o me lan process. x(t) = K cos (@, t+ 0) where A is a random variable over the ensemble and is uniformly distributed over the range (0, 2 n).3 Find the moment of the random variable X in terms of its characteristic function Фx (@)3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for tRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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