ECE 353 Probability and Random Signals - Practice Questions

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1 ECE 353 Probability and Random Signals - Practice Questions Winter 2018 Xiao Fu School of Electrical Engineering and Computer Science Oregon State Univeristy Note: Use this questions as supplementary materials to your homework and exam questions. The final exam questions are not necessarily with similar forms/patterns as the practice questions. 1 Basic Concepts, Set Theory Important concepts: experiments, elementary outcomes, sample space, events, probability axioms, conditional probability, law of total probability, independent trials, independence between events, Bayes Theorem, counting methods... Q1. Deer ticks can carry both Lyme disease and human granulocytic ehrlichiosis (HGE). In a study of ticks in the Midwest, it was found that 16% carried Lyme disease, 10% had HGE, and that 10% of the ticks that had either Lyme disease or HGE carried both diseases. (a) What is the probability P [L H] that a tick carries both Lyme disease (L) and HGE (H). (b) What is the conditional probability that a tick has HGE given that it has Lyme disease? Q2. Is it possible for A and B to be independent events yet satisfy A = B? Q3. Use a Venn diagram in which the area of the event proportional to its probability to illustrate two events A and B that are independent. Q4. In an experiment, A, B, C, and D are events with probabilities P [A B] = 5/8, P [A] = 3/8, P [C D] = 1/3, and P [C] = 1/2. Furthermore, A and B are disjoint, while C and D are independent. Find P [A B], P [B], P [A B c ], and P [A B c ]. Are A and B independent? Find P [D], P [C D c ], P [C c D c ], and P [C D]. Find P [C D] and P [C D c ]. Are C and D c independent? 1

2 rformer s birthday. All sion errors occasionally ACK) or a nonacknowlted back to the source to received packet. When, the packet is retransdelay sensitive and a maximum of d times. If independent Bernoulli y p, what is the PMF of acket is transmitted? L in the series. he number of calls necr? number of successes in n trials, n For a binomial random variable K representing the k=0 P K (k) = 1. Use this fact to prove the binomial theorem for any of finding the winner on 2 Discrete Random a > 0 andvariables b > 0. That is, show that hat the station will need n ( ) d a winner? (a + b) n n = a k b n k. k expectation, variance and standard deviation, conditional k=0 PMF. cations system, a source g digitized speech Q5. Discrete to a random Discrete variable Yrandom has thevariable CDF F Y (y) Y has shown: the CDF F Y (y) as shown: Important concepts: probability mass function, Bernoulli RV, Binomial RV, Geometric RV, Poisson RV, Cumulative distribution function, expectation, derived RV, fundamental theorem of y (a) Use the CDF to (a) findp[y the following < 1] probabilities: P [Y < 1], P [Y 2]. on day 1) you buy one (b) What is the PMF(b) ofp[y Y? 1] ity 1/2; otherwise, you winner with probability (c) What is F W (w)?(c) P[Y > 2] me of all other (d) tickets. Given B = Y (d) 3}. P[Y Find out 2] E[Y B]. day i you do not buy a (e) P[Y = 1] t that on day i, you buy the event that Q6. on Given day i the random(f) variable P[Y = Y3] in Q5, let U = g(y ) = Y 2. (a) Find P U (u). (g) P Y (y) 87], and P[N 99 ]? (b) Find F U (u) The random variable X has CDF he day on which you buy Find the PMF P(c) K (k). Find E U (u). 0 x < 1, x < 0, number of losing lottery F X (x) = x < 1, sed in m days. Q7. The binomial random variable X has PMF 1 x 1. he day on which you buy ( 5 ) hat is P D (d)? Hint: If (a) Draw P X (x) a graph = x (1/2) of the 5, x 0, 1, 2, 3, 4, 5}, CDF. F Y (y) Use the CDF to find the following probabilities: (a) Find the the expectation of X, i.e., µ X. (b) Find E[X 2 ]. (c) What is the standard deviation σ x? (d) Find the probability P [µ x σ x X µ x + σ x ]? 2

3 Q8. Every day you consider going jogging. Before each mile, including the first, you will quit with probability q, independent of the number of miles you have already run. However, you are sufficiently decisive that you never run a fraction of a mile. Also, we say you have run a marathon whenever you run at least 26 miles. (a) Let M equal the number of miles that you run on an arbitrary day. Find the PMF P M (m). (b) Let r be the probability that you run a marathon on an arbitrary day. Find r. (c) Define K = M 26. Let A be the event that you have run a marathon. Find P K A (k). 3 Continuous Random Variable Important concepts: probability density function, CDF, expectation, uniform RV, Exponential RV, Gaussian RV, standard Gaussian, Φ-function, Q-function, mixed RV, Derived RV, fundamental theorem of expectation, Conditioning a continuous RV Q9. For a uniform (0, 1) random variable U, find the CDF and PDF of Y = a + (b a)u with a < b. Show that Y is a uniform (a, b) random variable. Q10. The CDF of a RV is 0, x < 1 (1/2)(x + 1) F X (x) = 2, 1 < x 0 1 (1/2)(1 x) 2, 0 < x 1 1, x > 1 Sketch F X (x). Find the PDF f X (x) and sketch it. Find P [X 0]. Find P [ 1 2 < X 1 2 ]. Q11. W is a Gaussian random variable with expected value µ = 0, and variance σ 2 = 16. Given the event C = W > 0}, What is the conditional PDF, f W C (w)? Find the conditional expected value, E[W C]. Find the conditional variance, Var[W C]. Q12. X is a continuous random variable with CDF F X (x). Let Y = g(x) where 10, x < 0 g(x) =. 10, x 0 Express F Y (y) in terms of F X (x). 3

4 4 Pairs of Random Variables Important Concepts: Joint CDF, Joint PMF/PDF, marginal PMF/PDF, derived PDF from a pair, fundamental theorem of expectation, covariance, correlation, correlation coefficient, conditioning by an event, conditioning by a random variable, conditioned expected value, independent random variables, bivariate Gaussian RVs. Q13. Consider the two independent RVs X N (0, 1) and Y N (0, 1). Let Z = (X + Y ) 2. Find the mean of Z, E[Z]. Find r X,Z and r Y,Z. Determine if Z and Y are uncorrelated. Hint: E[(W µ) 3 ] = 0 for W N (µ, σ 2 ). Q14. Random variables X and Y have the joint PMF c x + y, x = 2, 0, 2, y = 1, 0, 1 P X,Y (x, y) = (a) What is the value of the constant c. (b) Given W = X + 2Y, what is P W (w)? (c) What is the expected value E[W ]? (d) What is the probability P [W > 0]? Q15. X and Y are independent random variables with PDFs: 2x, 0 x 1 3y 2, 0 y 1 f X (x) = f Y (y) = Let A = X > Y } (a) What are E[X] and E[Y ]? (b) What is E[X A]? (c) Repeat (b) if A = X Y }. Q16. Consider the two independent RVs X U[ 1, 1] and Y U[ 1, 1]. Let Z = X 2 Y. (a) Find the mean of Z, E[Z]. (10%) (b) Find Corr(X, Z) and Corr(Y, Z). (10%) 4

5 (c) Determine if Z and Y are uncorrelated. (5%) Q17. The joint PDF of X and Y is f X,Y (x, y) = Find the marginal PDFs f X (x) and f Y (y). 5y/4, 1 x 1, x 2 y 1 Q18. X 1 and X 2 are independent, identically distributed random variables with PDFs: x/2, 0 x 2 f X (x) = (a) Find the CDF, F X (x)? (b) What is P [X 1 1, X 2 1], the probability that X 1 and X 2 are both less than or equal to 1? (c) Let W = max(x 1, X 2 ). What is F W (1), the CDF of W evaluated at w = 1? (d) Find the CDF F W (w). 5 Random Vectors Homework is enough. 6 Sample Mean, Confidence Interval Important Concepts: Sample mean, expectation of sample mean, variance of sample mean, MGF, central limit theorem, Markov s inequality, Chebyshev s inequality, confidence level, confidence interval Q19. For an arbitrary random variable X, use the Chebyshev inequality to show that the probability that X is more than k standard deviations from its expected value E[X] satisfies P [ X E[X] 1/(kσ)] 1/k 2. For a Gaussian random variable Y, use the Φ-function to calculate the probability that Y is more than k standard deviations from its expected value E[Y ]. Compare the result to the upper bound based on the Chebyshev inequality. Q20. In n independent experimental trials, the relative frequency of event A is ˆP n (A). How large should n be to ensure that the confidence interval estimate has confidence coefficient 0.9? ˆP n (A) 0.05 P (A) ˆP n (A)

6 Q21. When we perform an experiment, event A occurs with probability P [A] = In this problem, we estimate P [A] using ˆP n (A), the relative frequency of A over n independent trials. How many trials n are needed so that the interval estimate has confidence coefficient 1 α = 0.99? ˆP n (A) P (A) ˆP n (A) How many trials n are needed so that the probability ˆP n (A) differs from P [A] by more than 0.1% is less than 0.01? Q22. Prove the Markov s inequality: Consider a nonnegative RV X (P [X < 0] = 0). Show that P [X c 2 ] E[X] c 2 Solution: where Therefore, we have P [X c 2 ] = u(x) = x=0 u(x c 2 )f X (x)dx, 1, x 0 P [X c 2 ] = 7 Random Process x=0 u(x c 2 )f X (x)dx x=0 (x/c 2 )f X (x)dx = E[X] c 2 Important Concepts: Random Process, Poisson Process, Expected Value Function, Autocorrelation, Stationary Process, Wide-Sense Stationary (WSS) Process Q23. Y (t) = A cos(2πf c t + θ); θ U[0, 2π] is random. Is Y (t) wide sense stationary? Solution: Let α(t) = 2πf c t In addition, we have E[Y (t)] = AE[cos(α(t) + θ)] = A = A 2π cos(α(t) + θ)dθ = 0. cos(α(t) + θ) 1 2π dθ R X (t, τ) = E[X(t)X(t + τ)] = A 2 E[cos(2πf c t + θ) cos(2πf c (t + τ) + θ)] 6

7 Recall that cos A cos B = 1 2 cos(a B) + 1 cos(a + B). 2 Hence, we have Therefore, Y (t) is indeed WSS. R X (t, τ) = A2 4π + A2 = A2 4π 4π cos(4πf c t + 2πf c τ + 2θ)dθ cos( 2πf c τ)dθ cos(2πf c τ)dθ = R X (0, τ). 7