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hw1 - EE 547 Homework 1 Due Tue Sep 9(1(4 Solve problems...

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EE 547: Homework 1 Due: Tue. Sep. 9 (1) – (4) Solve problems 1-3, 1-5, 1-8, 1-20 from Leon Garcia and Widjija’s book. Probability Review Problems (5) – (10) (5) Prove or disprove the following statement: (a) If random variables X and Y have identical distributions, then X = Y . (b) If random variables X and Y have identical distributions, then E [ X ] = E [ Y ]. (c) If X is independent from Y , and Y is independent from Z , then X is independent from Z . Note: In order to prove that a statement is true, you need to give a mathematical proof. In order to prove that a statement is false, you need to construct a counter-example. (6) Let X be a non-negative random variable with distribution F . Show that E ( X ) = integraldisplay 0 P ( { X > x } ) dx. and E ( X n ) = integraldisplay 0 nx n - 1 P ( { X > x } ) dx. (7) 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.
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