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# hw9 - Homework Set No 9 Probability Theory(235A Fall 2009...

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Homework Set No. 9 – Probability Theory (235A), Fall 2009 Posted: 11/24/09 — Due: Friday, 12/4/09 (Note extended due date!) 1. Compute the characteristic functions for the following distributions. (a) Poisson distribution: X Poisson( λ ). (b) Geometric distribution: X Geom( p ) (assume a geometric that starts at 1). (c) Uniform distribution: X U [ a, b ], and in particular X [ - 1 , 1] which is espe- cially symmetric and useful in applications. (d) Exponential distribution: X Exp( λ ). (e) Symmetrized exponential: A r.v. Z with density function f Z ( x ) = 1 2 e -| x | . Note that this is the distribution of the exponential distribution after being “symmetrized” in either of two ways: (i) We showed that if X, Y Exp(1) are independent then X - Y has density 1 2 e -| x | ; (ii) alternatively, it is the distribution of an “exponential variable with random sign”, namely ε · X where X Exp(1) and ε is a random sign (same as the coin flip distribution mentioned above) that is independent of X .

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hw9 - Homework Set No 9 Probability Theory(235A Fall 2009...

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