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160a09_hw3 - explanation for your result Problem 3 Problem...

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Pstat160a Winter 2009 Hw #3 Problem 1 Let ( X,Y ) be discrete random variables with joint probability generating function G X,Y ( s 1 ,s 2 ) = exp ± λ 1 ( s 1 - 1) + λ 2 ( s 2 - 1) + γ ( ( s 1 - 1)( s 2 - 1)( s 1 - s 2 ) Show, using a general result from a previous discussion section, that X and Y are both Poisson random variable as well as X + Y . Identify the parameters for each case. Argue that this distribution provides a counter example showing that the converse of the property for sum of independent Poisson random variable shown in class is not true. Problem 2 Use the rejection method to simulate an arbitrary geometric random variable from a a geometric (1/2). For what values values of p would the procedure work? What is the expected number of iteration and give an intuitive
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Unformatted text preview: explanation for your result. Problem 3 Problem 14, ch 11, only a) b) and c). Problem 4 a) Let X be any random variable, and a ∈ R fixed. Show that cov( X,a ) = 0. b) Let U ∼ U (0 , 1), and let V = 1-U . Show that cov( U,V ) < 0. Problem 5 Download the Matlab program ’rejectgeo.m’ from the class webpage, which implement the algorithm for problem 2 above. Explain it, run it for various values of p , check that indeed only some values of p work and comment. Explain what happens for the p that are not valid. Try also smaller and larger values for C (the bound for the ratio of pdf)....
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