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la. (5 pts) Show that A={,:n eN} is neither open nor closed; 1b. (15 pts) Let...
15 Let X and Y have a trinomial distribution with n = 8, P1 = 0.4...
15 Let X and Y have a trinomial distribution with n = 8, P1 = 0.4 and P2 = 0.1 f (x,y) = 8! 10.4*0.190.58---8,0 < x + y < 8,2 € N, Y EN x!y! (8 - x - y)! (a) Find E (Y|X = x), Var (Y|X = x) (b) Compute E (XY)
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}....
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
Let x[n] and y[n] be periodic signals with common period N, and let z[n] = {...
Let x[n] and y[n] be periodic signals with common period N, and let z[n] = { x[r]y[n – r) r=<N> be their period convolution. Let z[n] = sin(7") and y[n] = { . 0 <n<3 4 <n <7 Asns? be two signals that are periodic with period 8. Find the Fourier series representation for the periodic convolution of these signals.
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let...
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
1 4.6.3. (Harder!) Let 0 < a < 1. Prove that for any n EN, (1...
1 4.6.3. (Harder!) Let 0 < a < 1. Prove that for any n EN, (1 – a)” < 1+n·a
(15 pts) 7) Using the Divergence Theorem, calculate the flux integral JSF dĀ where F(x, y,...
(15 pts) 7) Using the Divergence Theorem, calculate the flux integral JSF dĀ where F(x, y, z) =< 2 + x2,r2 + y2y +> and S is the closed cylinder 22 + y2 = 1 with 03:31.
2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit...
2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit ball in f = {(21,..., In, ...): -10 < } is not compact. (Hint: find a sequence on the unit sphere containing no convergent subsequence)
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y...
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = -...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 :...
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
15 Let X and Y have a trinomial distribution with n = 8, P1 = 0.4 and P2 = 0.1 f (x,y) = 8! 10.4*0.190.58---8,0 < x + y < 8,2 € N, Y EN x!y! (8 - x - y)! (a) Find E (Y|X = x), Var (Y|X = x) (b) Compute E (XY)
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
Let x[n] and y[n] be periodic signals with common period N, and let z[n] = { x[r]y[n – r) r=<N> be their period convolution. Let z[n] = sin(7") and y[n] = { . 0 <n<3 4 <n <7 Asns? be two signals that are periodic with period 8. Find the Fourier series representation for the periodic convolution of these signals.
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
1 4.6.3. (Harder!) Let 0 < a < 1. Prove that for any n EN, (1 – a)” < 1+n·a
(15 pts) 7) Using the Divergence Theorem, calculate the flux integral JSF dĀ where F(x, y, z) =< 2 + x2,r2 + y2y +> and S is the closed cylinder 22 + y2 = 1 with 03:31.
2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit ball in f = {(21,..., In, ...): -10 < } is not compact. (Hint: find a sequence on the unit sphere containing no convergent subsequence)
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
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