E X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.

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1 E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator, Lecture notes (as in Fronter). 1

2 Problem 1 (10 points) Let A and B be two events with probabilities P (A) = a and P (B) = b, respectively, where a and b are real-valued constants 0 < a, b 1. Determine the conditional probability P (A B) if 1.1) A and B are mutually exclusive. 1.2) A and B are independent. 1.3) A B. 1.4) B A. 1.5) P (A B) = c, where c is a real-valued constant and 0 < c a. Problem 2 (10 points) An experiment is defined by flipping an unfair coin three times, where the result of each flip is independent of any preceding result. The probability of coming up heads h and tails t equals P (h) = 2/3 and P (t) = 1/3, respectively. 2.1) Determine all possible outcomes and their corresponding probabilities. 2.2) Find the probability P (A) of the event A that the first flip results in a head. 2.3) Find the probability P (B) of the event B that the number of heads is even. 2.4) Find the probability P (A B) of the events A and B. 2.5) Find the conditional probability P (A B) of the events A and B. 2

3 Problem 3 (15 points) The probability density function (PDF) f X (k) of a discrete random variable X is given by f X (k) = P {X = k} = a(k 2 + 4), k = 0, 1, 2, 3, 4, 0, otherwise, (3.1) where a is a real-valued constant. 3.1) Determine the constant a and sketch the PDF f X ( k) of X. 3.2) Find and sketch the corresponding cumulative distribution function (CDF) F X (k) of X. 3.3) Find the probability P {X > 1}. 3.4) Find the conditional probability P {X = 3 X 2}. 3.5) Find the mean E{X} of X. 3.6) Find the variance Var{X} of X. Problem 4 (10 points) Let X be a uniformly distributed random variable over the interval [1, 3]. Given is another random variable Y, which is defined as Y = 6 X ) Sketch the PDF f X (x) of X. 4.2) Find the PDF f Y (y) of Y. 4.3) Sketch the PDF f Y (y) of Y. 3

4 Problem 5 (20 points) Let X and Y be two random variables, which are characterized by the joint PDF f XY (x, y) = xy, if 0 x 1 and 0 y 2, 0, otherwise. (5.1) 5.1) Find the marginal PDF f X (x) of X. 5.2) Find the marginal PDF f Y (y) of Y. 5.3) Are X and Y dependent or independent? Explain your answer. 5.4) Are X and Y correlated or uncorrelated? Explain your answer. 5.5) Find the covariance C XY of the two random variables X and Y. 5.6) Find the probability P {X 2 + Y 2 1}. Hints: 1. The Cartesian coordinates (x, y) can be transformed into polar coordinates (r, θ) by means of x = r cos(θ), y = r sin(θ), and dx dy = r dr dθ. 2. sin(α) cos(β) = 1 2 [sin(α + β) + sin(α β)] 3. sin(ax)dx = 1 a cos(ax) 4

5 Problem 6 (15 points) Let X i (i = 1, 2,..., n) be the lifetime of a light bulb, which is used until it fails, and then it is replaced by a new light bulb. The lifetimes X i of the light bulbs are independent and identically distributed (i.i.d.) random variables with mean E{X i } = µ X = 100 h (in hours) and variance Var{X i } = σ 2 X = 25 h 2. Another random variable Y is defined as the total lifetime of n light bulbs, which are used one-by-one, i.e., Y = X 1 + X X n. 6.1) Find the mean E{Y } and the variance Var{Y } of Y. 6.2) Give reasons, why the cumulative distribution function (CDF) F Y (y) of Y can be expressed by F Y (y) = G y nµ X nσ 2 X if n where G( ) is the CDF of the standard normal random variable. 6.3) For n = 36, find an approximate expression for the probability that Y is between 3500 and 3700, i.e., P {3500 Y 3700}. Hint: Some useful values of the CDF G(x) of the standard normal random variable: G(1) = 0.6 G(3) = 0.7 G(10/3) =

6 Problem 7 (20 points) Let X(t) = e At be a stochastic process, where A is a real-valued random variable, which is uniformly distributed between 1 and ) Compute the mean µ X (t) of X(t) by using µ X (t) = E{X(t)}. 7.2) Compute the autocorrelation function R XX (t 1, t 2 ) of X(t) by using R XX (t 1, t 2 ) = E{X(t 1 )X (t 2 )}. 7.3) Is the stochastic process X(t) wide-sense stationary (WSS)? Give reasons for your answer. 7.4) Compute the time average µ X of a single sample function X(t; a i ) using µ X =< X(t; a i ) >, where a i is a constant (outcome of A) and < > denotes the time averaging operator. 7.5) Is the stochastic process X(t) mean-ergodic? Give reasons for your answer. 7.6) Compute the autocorrelation function R XX (τ) of a single sample function X(t; a i ) using R XX (τ) =< X(t + τ; a i )X (t; a i ) >. 7.7) Is the stochastic process X(t) autocorrelation ergodic? Give reasons for your answer. Hint: e ax dx = eax, where a 0 is a constant. (7.1) a 6