Final. Fall 2016 (Dec 16, 2016) Please copy and write the following statement:

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1 ECE 30: Probabilistic Methods in Electrical and Computer Engineering Fall 06 Instructor: Prof. Stanley H. Chan Final Fall 06 (Dec 6, 06) Name: PUID: Please copy and write the following statement: I certify that I have neither given nor received unauthorized aid on this exam. (Please copy and write the above statement.) (Signature) Problem Score Q Q Q3 Q4 Q5 Total c 06 Stanley Chan. All Rights Reserved.

2 Problem. (0 points) Determine whether the following statements are (A statement is true if it is always true. Otherwise it is false.) Circle your answer. No partial credit will be given. (a) Let A, B, X be three events, and assume that A B = Ω. Then, P[X] = P[X A\B] + P[X A B] + P[X B\A]. (b) For any random variable X, the variance has the property that Var[aX + b] = avar[x] + b. (c) Let X 0 and Y 0 be two independent random variables. Then, P[X + Y ] P[X Y ]. (d) Let X Uniform[, ]. Then F X (x) = x +. (e) Let X be a random variable with a PDF Then, c =. f X (x) = { π exp { x }, x < 0, cλ exp{ λx}, x 0. c 06 Stanley Chan. All Rights Reserved.

3 (f) The probability P[X = b] is determined by P[X = b] = lim h 0 F X (b + h) F X (b). (g) Let X be a random variable with CDF F X (x). Let Y = X + 3. Then, ( ) y 3 F Y (y) = F X. (h) Let M N = N ( ) N n= X n be the sample mean of a sequence of random variables X,..., X N. If M N N µ, σ N, then ( ) MN µ N N (0, ). σ (i) Let X, Y, Z be three random variables. It is given that X and Y are uncorrelated. Let a be a constant. Then, Cov(X, Y + az) = acov(x, Z). (j) Let X, Y be two independent random variables such that X Bernoulli(p) and Y Exponential(α). Let U = 3XY. Then, the moment generating function of U is ( ) 3α M U (s) = p + p. α s c 06 Stanley Chan. All Rights Reserved. 3

4 Problem. (0 points) Circle one and only one answer. If I cannot tell which answer you are circling, I will give you a zero. There is no partial credit for this problem.. Consider a pair of random variables X and Y with joint PMF given by the following table. Y= 3 4 X = The conditional expectation E[X {Y = Y = }] is (a) (b) (c) 7 3 (d) 0 (e) 9 (f) 5 (g) 7 (h) 4 3 (i) None of the above (j) Problem undefined Let X Exponential(λ). Let Y be a random variable such that { X c, X c, Y = 0, X < c. Then, the PDF of Y is (a) f Y (0) = 0 and f Y (y) = λe λy for y > 0. (b) f Y (0) = and f Y (y) = λe λy for y > 0. (c) f Y (0) = e λc and f Y (y) = λe λy for y > 0. (d) f Y (0) = e λc and f Y (y) = λe λy for y > 0. (e) f Y (0) = e λc and f Y (y) = λe λ(y c) for y > 0. (f) f Y (0) = e λc and f Y (y) = λe λ(y+c) for y > 0. (g) f Y (0) = e λc and f Y (y) = λe λ(y c) for y > 0. (h) f Y (0) = e λc and f Y (y) = λe λ(y+c) for y > 0. (i) None of the above. (j) Problem undefined. c 06 Stanley Chan. All Rights Reserved. 4

5 3. Let X and Y be two independent random variables with PDF f X (x) and f Y (y), respectively. Let Z = X + Y. Then, F Z (z) = (a) (b) (c) (d) (e) z y y z y+z z x x z (f) (g) (a) and (d) (h) (b) and (e) (i) (c) and (f) f X(x)f Y (y)dxdy f X(x)f Y (y)dxdy f X(x)f Y (y)dxdy f X(x)f Y (y)dydx f X(x)f Y (y)dydx x+z f X(x)f Y (y)dydx (j) None of the above 4. Let Θ be a random parameter with prior distribution Θ N (0, ). Let X be a Gaussian random variable with mean Θ and variance σ. Suppose that we have observed two realizations x and x. The maximum-a-posteriori (MAP) estimate of Θ is (a) θ MAP = x+x (b) θ MAP = x+x σ (c) θ MAP = x σ + x (d) θ MAP = x + x σ (e) θ MAP = xσ +x (f) θ MAP = x+xσ (g) θ MAP = σ (x +x ) (h) θ MAP = x+x +σ (i) θ MAP = σ x +x +σ (j) None of the above. 5. Let X(t) be a WSS process with power spectral density S X (ω). Let Y (t) = X(t) X(t d) for some positive constant d. The power spectral density S Y (ω) is (a) S X (ω)( + cos(ωd)) (b) S X (ω)( cos(ωd)) (c) S X (ω) cos(ωd) (d) S X (ω)( + sin(ωd)) (e) S X (ω)( sin(ωd)) (f) S X (ω)( e jωd e jωd ) (g) S X (ω)( + e jωd ) (h) S X (ω)( e jωd ) (i) S X (ω)( + e ωd ) (j) None of the above. c 06 Stanley Chan. All Rights Reserved. 5

6 Problem 3. (0 points) Let X and Y have a joint PDF for 0 x and 0 y. f X,Y (x, y) = c(x + y), (a) (3 points) Find c. (b) (4 points) Find f X (x) and f Y (y). (c) (3 points) Find E[ ]. X+ c 06 Stanley Chan. All Rights Reserved. 6

7 (d) (3 points) Find f Y X (y x). (e) (3 points) Find E[Y X = x]. (f) (4 points) Find P[Y > X X > ]. c 06 Stanley Chan. All Rights Reserved. 7

8 Problem 4. (0 points) Let θ be an unknown scalar. Let W n N (0, σ ) be an i.i.d. Gaussian noise. Let Y n = θ + W n, for n =,..., N. Suppose that we have observed realizations y,..., y N for Y,..., Y N. (a) (5 points) Find the joint PDF of Y,..., Y N given the parameter θ, i.e., f Y,...,Y N (y,..., y N θ). (b) (5 points) Recall that the maximum likelihood estimate θ ML is defined as θ ML = argmax θ f Y,...,Y N (y,..., y N θ) Find θ ML. Express your answer in terms of y,..., y N. c 06 Stanley Chan. All Rights Reserved. 8

9 (c) (5 points) Let M N = N N n= Y n. Find the moment generating function of M N. (d) (5 points) Use central limit theorem, find the probability Express your answer in terms of the Φ( ) function. P[ M N θ σ]. c 06 Stanley Chan. All Rights Reserved. 9

10 Problem 5. (0 points) Let X(t) be a WSS process with autocorrelation function R X (τ) and power spectral density S X (ω) as shown below. π W S X (ω) W W Let Θ Uniform[0, π] be a uniformly distributed random variable. Define (a) (5 points) Find E[Y (t + τ)y (t)], i.e., R Y (τ). Y (t) = X(t) cos(ω 0 t + Θ). c 06 Stanley Chan. All Rights Reserved. 0

11 (b) (5 points) Find S Y (ω). Express your answer in terms of S X (ω). (Hint: f(t)e jω0t F (ω ω 0 ).) Now, assume that Y (t) is passed through an LTI system shown in the following block diagram. Y (t) U(t) Z(t) H(ω) cos(ω0 t + Θ) (c) (5 points) Find and sketch S U (ω). Express your answer in terms of S X (ω). Mark and label your figure clearly. c 06 Stanley Chan. All Rights Reserved.

12 (d) (5 points) Assume that H(ω) is given by Find R Z (τ). H(ω) = {, W ω W, 0, otherwise. c 06 Stanley Chan. All Rights Reserved.

13 Useful Identities. r k = + r + r +... = r 4. k=0 k= n n. k = n = n(n+) 5. k= k= 3. e x = x k k! = + x! + x! (a + b)n = n k=0 Common Distributions kr k = + r + 3r +... = ( r) k = n = n3 3 + n + n 6 k=0 ( n ) k a k b n k Bernoulli P[X = ] = p E[X] = p Var[X] = p( p) M X (s) = p + pe s Binomial p X (k) = ( ) n k p k ( p) n k E[X] = np Var[X] = np( p) M X (s) = ( p + pe s ) n Geometric p X (k) = p( p) k E[X] = p Var[X] = p p M X (s) = Poisson pe s ( p)e s p X (k) = λk e λ k! E[X] = λ Var[X] = λ M X (s) = e λ(es ) Gaussian f X (x) = (x µ) e σ πσ E[X] = µ Var[X] = σ M X (s) = e µs+ σ s Exponential f X (x) = λ exp { λx} E[X] = λ Var[X] = λ M X (s) = λ λ s Uniform f X (x) = b a E[X] = a+b Var[X] = (b a) M X (s) = esb e sa s(b a) Fourier Transform Table f(t) F (w) f(t) F (w). e at u(t) a+jw, a > 0 0. sinc ( W t π ) W ( w W ). e at u( t) a jw, a > 0. e at w sin(w 0 t)u(t) 0, a > 0 (a+jw) +w0 3. e a t a a +w, a > 0. e at cos(w 0 t)u(t) a+jw, a > 0 (a+jw) +w 0 a 4. a +t πae a w, a > 0 3. e t σ πσe σ w 5. te at u(t) (a+jw), a > 0 4. δ(t) 6. t n e at u(t) n! (a+jw) n+, a > 0 5. πδ(w) 7. rect( t wτ τ ) τsinc( ) 6. δ(t t 0) e jwt0 8. sinc(w t) π W rect( w W ) 7. ejw0t πδ(w w 0 ) 9. ( t τ ) τ sinc ( wτ 4 ) Some definitions: sinc(t) = sin(t) t Basic Trigonometry rect(t) = {, 0.5 t 0.5, 0, otherwise. (t) = e jθ = cos θ + j sin θ, sin θ = sin θ cos θ, cos θ = cos θ. { t, 0.5 t 0.5, 0, otherwise. cos A cos B = (cos(a + B) + cos(a B)) sin sin A cos B = (sin(a + B) + sin(a B)) cos A sin B = (cos(a + B) cos(a B)) A sin B = (sin(a + B) sin(a B)) c 06 Stanley Chan. All Rights Reserved. 3