# rec05 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 5 September 23, 2010 1. (a) Derive the expected value rule for functions of random variables E [ g ( X )] = x g ( x ) p X ( x ). (b) Derive the property for the mean and variance of a linear function of a random variable Y = aX + b . E [ Y ] = a E [ X ] + b, var( Y ) = a 2 var( X ) . (c) Derive var( X ) = E [ X 2 ] - ( E [ X ]) 2 2. A marksman takes 10 shots at a target and has probability 0.2 of hitting the target with each shot, independently of all other shots. Let X be the number of hits. (a) Calculate and sketch the PMF of X . (b) What is the probability of scoring no hits? (c) What is the probability of scoring more hits than misses? (d) Find the expectation and the variance of X . (e) Suppose the marksman has to pay \$3 to enter the shooting range and he gets \$2 dollars for each hit. Let Y be his pro±t. Find the expectation and the variance of

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rec05 - Massachusetts Institute of Technology Department of...

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