## Problem Set 1

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## Problem Set 1

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- Pages:
- 7
- School:
- Massachusetts Institute of Technology
- Course:
- 6 453 - Quantum Optical Communication

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MIT OpenCourseWare http ocw mit edu 6 453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 453 Quantum Optical Communication Problem Set 1 Fall 2008 Issued Thursday September 4 2008 Due Thursday September 11 2008 Reading For probability review Chapter 3 of J H Shapiro Optical Progagation Detection and Communication For linear algebra review Section 2 1 of M A Nielsen and I L Chuang Quantum Computation and Quantum Information Problem 1 1 Here we shall verify the elementary properties of the 1 D Gaussian probability density function pdf 2 2 e X m 2 px X 2 2 a By converting from rectangular to polar coordinates using X m R cos and Y m R sin show that 2 2 2 2 2 X m 2 2 2 dX e dX dY e X m 2 Y m 2 2 2 thus verifying the normalization constant for the Gaussian pdf b By completing the square in the exponent within the integrand ejvX X m dX 2 2 2 2 2 verify that Mx jv ejvm v 2 2 2 is the characteristic function associated with the Gaussian pdf c Di erentiate Mx jv to verify that E x m di erentiate once more to verify that var x 2 Problem 1 2 Here we shall verify the elementary properties of the Poisson probability mass function pmf mn m Px n e for n 0 1 2 and m 0 n 1 a Use the power series z e zn n 0 n to verify that the Poisson pmf is properly normalized b Use the power series for ez to verify that Mx jv exp m ejv 1 is the characteristic function associated with the Poisson pmf c Di erentiate Mx jv to verify that E x m di erentiate once more to verify that var x m Problem 1 3 Let x be a Rayleigh random variable i e x has pdf X e X 2 2 2 for X 0 2 px X 0 otherwise and let y x2 a Find py Y the pdf of y b Find my and y2 the mean and variance of the random variable y Problem 1 4 Let x and y be statistically independent identically distributed zero mean variance 2 Gaussian random variables i e the

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