ESE 520 Fall 2015Some practice problems for the final exam1. Are the following statements correct? A simple YES or NO will do, but read the statementscarefully.(a) Every subset of N is a Borel set.(b) Every subset of R is a Borel set.(c) Uncorrelated Gaussian random variables are independent.(d) The distribution of a large number of identically distributed random variables is approximatelyGaussian.(e) The distribution of a large number of independent exponential random variables is approxi-mately Gauss ian.(f) A counting proc e ss with stationary and independent increments that does not have explosionsis the Poisson process.(g) The covariance function of a stochastic process is po sitive definite.(h) White noise is a Gaussian stochastic process.2. Define the concepts which make up a probability space (Ω, F, P ) and explain what it means for amap X : Ω → R to be a random variable. Give an ex ample of a probability space (Ω, F, P ) and amap X that is NOT a random variable.3. Let X be a continuous random variable with distribution function F and density f . Define a newrandom variable Y as Y = F (X). Show that Y is uniformly distributed in the interval [0, 1].4. let X be a no n-negative continuo us random variable with mean µ. Show thatµ =Z∞0P (X ≥ t)dt.Hint: Express P (X ≥ t) in terms of the de nsity f of X.5. Let X and Y be independent scalar Gaussian r andom variables with mean 0 and variance 1. Cal-culate the distributions of the random variablesZ1=X2X2+ Y2, Z2=XYX2+ Y2, and Z3=Y2X2+ Y2,and find their expectations and variances.6. Let X and Y be independent scalar Gaussian random variables with mean 0 and variance 1 and letR and Θ denote the corresponding polar coordinates, i.e. X = R cos Θ a nd Y = R s in Θ. Computethe conditional expectations E[X2|Θ], E[XY |Θ] and E[Y2|Θ].7. The characteristic function of a χ2random variable with ν degrees of free dom is given byΦ(t) =11 − 2itν2.Show that, if Xi, i = 1, . . . , n are independent χ2random variables with νidegrees of freedom, thenthe sumZ = X1+ · · · + Xnis a χ2random variable with ν1+ · · · + νndegrees of freedom.8. Let (X, Y, Z) be a jointly continuous random vector w ith mean µ and covariance matrix R g ivenbyµ =113and R =4 −1 0−1 9 10 1 2.Measurements of the random variables Y and Z give the values y = 2 and z = 1. What is the bestlinear estimate for the value of X given these measurements?9. Let Ntbe a Poisso n process with intensity λ > 0. Compute its covariance function.10. Let A and B be independent Gaussian random variables with mean 0 and variance 1 and defineXt= R cos(2πt + φ)whereR =pA2+ B2and φ = arctanBA.Show that Xtis a stationary Ga ussian proc ess and calculate its mean µ and covariance function R.11. Let Xt, t ≥ 0 , be a wide-sense stationary stochastic process w ith zero mean and covariance functionRX(τ). Let Θ be a r andom variable that is indepe ndent of the process Xt, t ≥ 0, and uniformlydistributed over the interval [0, 2π]. Define a new stochastic process Yt, t ≥ 0, byYt= Xtcos(2πt + Θ).Compute the mean µ(t) = E[Yt] and covariance function RY(s, t) = E[(Ys− µ(s))(Yt− µ(t))] of theprocess Yt. Is Ytwide-sense stationary?12. Let Wt, t ≥ 0, be a standard Wiener proces s. The refle c ted Brownian motion is the stochasticprocess defined by Xt= |Wt|. Compute the mean and variance of Xt.13. Let Wt= W (t), t ≥ 0, be a standard (one-dimensional) Wiener process. Define a new stochasticprocess Xt= X(t), t > 0, asX(t) = tW1tand set X(0) = 0. Using the strong law of large numb e rs it can be shown (and you do NOT needto prove this) thatlimt→∞tW1t= 0.Use this fact to show that Xtalso is a standa rd Wiener process.14. Let Xt, t ∈ R, be a wide-sense stationary stochastic process with zero mean a nd covariance functionRX(τ) = exp−12τ2. What is its spectr al densitySX(ν) =Z∞−∞RX(τ) exp (−2πiντ) dτ?Hint: What is the characteristic function of a Gauss ian random variable w ith mean µ and