rec06 - (1/8) (1/8) (1/8) (1/8) (1/8) (1/8) (1/8) (1/8) y x...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 6 September 28, 2010 1. Consider an experiment in which a fair four-sided die (with faces labeled 0, 1, 2, 3) is thrown once to determine how many times a fair coin is to be Fipped. In the sample space of this experiment, random variables N and K are de±ned by N = the result of the die roll K = the total number of heads resulting from the coin Fips (a) Determine and sketch p N ( n ) (b) Determine and tabulate p N,K ( n,k ) (c) Determine and sketch p K | N ( k | 2) (d) Determine and sketch p N | K ( n | 2) 2. Consider an outcome space comprising eight equally likely event points, as shown below:
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Unformatted text preview: (1/8) (1/8) (1/8) (1/8) (1/8) (1/8) (1/8) (1/8) y x 1 2 3 4 1 2 3 (a) Which value(s) of x maximize(s) E [ Y | X = x ]? (b) Which value(s) of y maximize(s) var( X | Y = y )? (c) Let R = min( X,Y ). Prepare a neat, fully labeled sketch of p R ( r ), (d) Let A denote the event X 2 Y . Determine numerical values for the quantities E [ XY ] and E [ XY | A ]. 3. Example 2.17. Variance of the geometric distribution. You write a software program over and over, and each time there is probability p that it works correctly, independent of previous attempts. What is the variance of X , the number of tries until the program works correctly? Page 1 of 1...
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