hw1 - CSE-ECE 861 Homework 1 Due in class by Wed Feb 1...

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CSE-ECE 861: Homework 1 Due in class by: Wed. Feb. 1, 2012 (1) – (4) Solve problems 1.2, 1.5, 1.9, 1.15 from Leon Garcia and Widjija’s book (Second Edition). Probability Review Problems (5) – (11) (5) Prove that finite additivity follows from countable additivity (6) From the axioms of probability, prove the following: For an event A and its complement A c , prove that P ( A c ) = 1 - P ( A ) If A B , then P ( A ) P ( B ). (7) Let X be a non-negative random variable with distribution F . Show that E ( X ) = Z 0 P ( { X > x } ) dx. and E ( X n ) = Z 0 nx n - 1 P ( { X > x } ) dx. (8) X and Y are random variables. (a) Show that E ( X ) = E ( E ( X | Y )). (b) If P ( { X x, Y y } ) = P ( { X x } ) P ( { Y y } ) then show that E ( XY ) = E ( X ) E ( Y ), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove. (c) The moment generating function of a random variable Z is defined as Ψ Z ( t ) := E ( e tZ ). Now if X and Y are independent random variables then show that Ψ X + Y ( t ) = Ψ X ( t Y ( t ) .
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