FINAL EXAM: 3:30-5:30pm

Transcription

1 ECE 30: Probabilistic Methods in Electrical and Computer Engineering Spring 016 Instructor: Prof. A. R. Reibman FINAL EXAM: 3:30-5:30pm Spring 016, MWF 1:30-1:0pm (May 6, 016) This is a closed book exam. There are 1 problems. Neither calculators nor help sheets are allowed. Some formulae and helpful information is attached at the end. Cheating will result in a zero on the exam and possibly failure of the class. Do not cheat! Use of any electronics is considered cheating. Put your name on every page of the exam and turn in everything when time is up. Name: PUID: I certify that I have neither given nor received unauthorized aid on this exam. Signature:

2 Problem 1. (Multiple choice: 5 points) Two fair dice are rolled. Let X be the absolute value of the difference between the numbers on each die. What is P (X 3)? (a) 7/9 (b) /3 (c) 5/9 (d) 4/9 (e) 1/3 Problem. (5 points) X and Y are independent random variables, with moment generating functions M X (s) = (1 s) 3 and M Y (s) = (1 s). What is the second moment of Z = X + Y?

3 Problem 3. (5 points) On Tuesday next week, the probability that a Purdue student exercises is 0.7, while the probability a student eats broccoli is 0.. Of those students who exercise next Tuesday, 15% will eat broccoli. What is the probability that a randomly selected student doesn t exercise and doesn t eat broccoli next Tuesday. 3

4 Problem 4. (Multiple choice: 5 points) Let Z = 3X Y 5, where X and Y are independent random variables with V ar(x) = 1 and V ar(y ) =. What is V ar(z)? (a) 4 (b) 7 (c) 11 (d) 16 (e) None of the above. Problem 5. (Multiple choice: 5 points) Suppose X(t) is a WSS random process with mean m X = 0 and autocorrelation function R X (τ) = exp( τ ). What is the correlation coefficient between X() and X( 1)? (a) e 3 (b) 0 (c) e e 1 (d) e 6 (e) None of the above. 4

5 Problem 6. (5 points) A super-computer has three cooling components that operate independently. Each fails with probability 1/10. The super-computer will overheat if any two (or three) cooling components fail. What is the probability the super-computer overheats? Problem 7. (Multiple choice: 5 points) The lifetime of a machine, X, is a Gaussian random variable with mean 10 and variance 4. What is the value of x for which the machine has an 1% chance of surviving x or more years? (You may use (and detach) the Φ-function table on the last page of the exam. If you draw a clear picture you may get partial credit.) (a) 5.30 (b) 7.65 (c) 8.41 (d) 1.35 (e)

6 Problem 8. (5 points) A device is deployed in a remote region. The time, T, to failure, is exponentially distributed with mean 3 years. The device will not be monitored during the first years, so the time before failure can be discovered is X = max(t, ). What is E(X)? (Hint: Draw a sketch of the PDF of X. If you can break it into parts and apply principles we learned in class, only one integration is necessary. If not, you may find the following integral (without limits) helpful.) ( x xe ax dx = a 1 ) a e ax 6

7 Problem 9. (5 points) Given the CDF of X, what is the PDF of Y = X, f Y (y). F X (x) = { 1 (/x) for x > 0 otherwise 7

8 Problem 10. (0 points) Suppose you build a system with a part that comes from either Company 1 (with probability /3) or Company (with probability 1/3). Let N indicate the company, so that N is a discrete RV with PMF /3 for n = 1 p N (n) = 1/3 for n = 0 otherwise Let X be the lifetime of the part, which has a different distribution whether the part comes from Company 1 or Company. In particular, let X be a continuous RV such that when N = 1, X is exponentially distributed with mean 5, and when N =, X is exponentially distributed with mean 8. (a) What is the conditional PDF of X given N=? (In other words, what is the PDF of X when the part comes from Company?) (b) Find the marginal PDF of X. (c) What is P (X > 8)? (d) If your device has lasted long enough that X > 8 already, what is the probability it is from Company 1, i.e., that N = 1? 8

9 Problem 11. (0 points) Let X be a continuous uniform RV on the interval [0, 1]. Conditioned on X, then Y is a continuous RV that is uniformly distributed on the interval [x, x + 1]. (a) What is P (Y > 0.5)? (Hint, draw a clear diagram of the region of support for (X, Y ).) (b) Find the marginal PDF of Y. (c) Find E(Y ). (Hint: are you able to use symmetry?) (d) Find COV (X, Y ). 9

10 Problem 1. (15 points) Suppose X n is a discrete-time random process with E(X n ) = 0 and VAR(X n ) = 3 for all n, where E(X i X j ) = 0 for i j. This random process is input to a digital filter to create Y n, where (a) Find E(Y n ). Y n = X n X n 1 for all n. (b) Find the auto-correlation of Y n, which is denoted R Y (m, k). (c) Is Y n a wide-sense stationary random process? Explain your answer by providing justification. 10

11 Empty page to show more work. Label problems clearly! (I need to be able to find your work to give you credit.) 11

12 Discrete Random Variables Bernoulli Random Variable, parameter p S = {0, 1} p 0 = 1 p, p 1 = p; 0 p 1 E(X) = p; VAR(X) = p(1 p) M X (s) = 1 + p + p exp(s) Binomial Random Variable, parameters (n, p) S = {0, 1,..., n} p k = ( n k) p k (1 p) n k ; k = 0, 1,..., n; 0 p 1 E(X) = np; VAR(X) = np(1 p) M X (s) = (1 + p + p exp(s)) n Geometric Random Variable, parameter p S = {0, 1,...} p k = p(1 p) k 1 ; k = 0, 1,..., ; 0 p 1 E(X) = (1 p)/p; VAR(X) = (1 p)/p M X (s) = pe s /(1 (1 p)e s ) Poisson Random Variable, parameter α S = {0, 1,...} p k = α k e α /k! k = 0, 1,..., E(X) = α; VAR(X) = α M X (s) = exp(α(e s 1)) Uniform Random Variable S = {0, 1,..., L} p k = 1/L k = 0, 1,..., L E(X) = (L + 1)/; VAR(X) = (L 1)/1 M X (s) = (1 exp(s(l + 1)))/(1 exp(s)) Continuous Random Variables Uniform Random Variable Equally likely outcomes S = [a, b] f X (x) = 1/(b a), a x b E(X) = (a + b)/; VAR(X) = (b a) /1 M X (s) = (exp(bs) exp(as))/(s(b a)) Exponential Random Variable, parameter λ S = [0, ) f X (x) = λ exp( λx), x 0, λ > 0 E(X) = 1/λ; VAR(X) = 1/λ M X (s) = λ/(λ s) One Gaussian Random Variable, parameters µ, σ S = (, ) f X (x) = exp( (x µ) /(σ ))/ πσ E(X) = µ; VAR(X) = σ M X (s) = exp(µs + s σ /) 1

13 Two Joint Gaussian Random Variables, parameters m X, σ X and m Y, σ Y S X = (, ), S Y = (, ) 1 f X,Y (x, y) = πσ X σ Y 1 ρ XY [ ( (x ) 1 mx exp (1 ρ XY ) ρ XY σ X ( ) ( x mx y my σ X σ Y ) ( ) )] y my + σ Y E(X) = m X ; E(Y ) = m Y ; VAR(X) = σx VAR(Y ) = σy Trigonometric identities Other useful formulas exp(jθ) = cos θ + j sin θ sin θ = sin θ cos θ; cos θ = cos θ 1 cos A cos B = 1 (cos(a + B) + cos(a B)) sin A cos B = 1 (sin(a + B) + sin(a B)) cos A sin B = 1 (sin(a + B) sin(a B)) sin A sin B = 1 (cos(a B) cos(a + B)) k=0 n k=0 kr k 1 = k=1 r k = 1 rn+1 1 r r k = 1 1 r n k=1 n k=0 if r < 1 1 (1 r) if r < 1 n k = k=1 n(n + 1) k = n3 3 + n + n 6 k=0 x k k! = ex ( ) n a k b n k = (a + b) n k 13

14 Fourier transform pairs and properties h(t) H(f) δ(t) 1 rect(t/τ) τ sinc(fτ) h(t a) exp(jπfa)h(f) when a is constant h(at) H(f/a)/ a exp( at ) a a + (πf ) g(t) h(t) G(f)H(f) g(t)h(t) G(f) H(f) dh(t)/dt jπf H(f) Definitions consistent with above pairs: sinc(t) = sin πt πt { 1 for 0.5 t 0.5 rect(t) = 0 otherwise 14

15 Table 1: Table of the Standard Normal Cumulative Distribution Function Φ(z) z