midterm_s2

midterm_s2 - EE 376A/Stat 376A Prof. T. Weissman...

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EE 376A/Stat 376A Information Theory Prof. T. Weissman Thursday, February 8, 2007 Midterm 1. ( 35 points ) Throwing a die Suppose you have a die with three sides. The probability of outcome of each side is given as X = 1 , w.p. 1 / 2 2 , w.p. 1 / 3 3 , w.p. 1 / 6 You throw this die independently n times, and collect the sequence of numbers ( X 1 ,X 2 , ··· ,X n ). (a) ( 15 points ) Suppose you are building an n dimensional rectangular box with side lengths R n = ( X 1 ,X 2 , ··· ,X n ). The volume of the box is V ( R n ) = Q n i =1 X i . The n -dimensional cube with “effective side length” is a cube with the same volume as the box, namely V ( R n ). When n is large, what is the effective side length ` for the rectangular box? More precisely, identify the value of ` for which ( V ( R n )) 1 /n ` in probability as n → ∞ . (b) ( 20 points ) Now, let P ( R n ) be the probability of box R n . Identify the value of p for which ( P ( R n ) 1 /n ) p in probability as n → ∞ . Interpret your result. ( Hint for both parts: You may want to evaluate the logarithm of the limit.) Solution: Throwing a die (a) Let’s take logarithm and get the limit, first. 1 n log V ( R n ) = 1 n log ± n Y i =1 X i ! = 1 n n X i =1 log X i E (log X i ) in probability where the convergence follows from the weak law of large numbers, and X i ’s being i.i.d. Since E (log X i ) = 1 2 log 1 + 1 3 log 2 + 1 6 log 3 = 1 3 + 1 6 log 3 , 1
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midterm_s2 - EE 376A/Stat 376A Prof. T. Weissman...

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