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hw3-sol - AMS 311(Fall 2010 Joe Mitchell PROBABILITY THEORY...

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AMS 311 (Fall, 2010) Joe Mitchell PROBABILITY THEORY Homework Set # 3 – Solution Notes (1). (30 points) In your pocket, you have 2 dimes, 2 nickels, and 1 penny. You select 3 coins at random (without replacement). Let X represent the amount (in cents) that you select from your pocket. (a). Give (explicitly) the probability mass function for X . Also show a plot of it. By considering each possible outcome, we see that the possible values for X are { 11 , 16 , 20 , 21 , 25 } . The probability function is given by p ( x ) = P ( X = x ) = ( 2 0 )( 2 2 )( 1 1 ) ( 5 2 ) = 0 . 1 if x = 11 ( 2 1 )( 2 1 )( 1 1 ) ( 5 2 ) = 0 . 4 if x = 16 ( 2 1 )( 2 2 )( 1 0 ) ( 5 2 ) = 0 . 2 if x = 20 ( 2 2 )( 2 0 )( 1 1 ) ( 5 2 ) = 0 . 1 if x = 21 ( 2 2 )( 2 1 )( 1 0 ) ( 5 2 ) = 0 . 2 if x = 25 0 otherwise The plot of p ( x ) simply consists of isolated dots of the appropriate heights at x = 11 , 16 , 20 , 21 , 25, with p ( x ) = 0 everywhere else. x p(x) 1.0 11 20 21 25 16 0.5 (b). Give (explicitly) the cdf, F ( x ) , for X . Also show a plot of it. The cdf is a staircase with step heights given by the spike heights of the pmf: F ( x ) = P ( X x ) = 0 if x < 11 0 . 1 if 11 x < 16 0 . 5 if 16 x < 20 0 . 7 if 20 x < 21 0 . 8 if 21 x < 25 1 if x 25

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x F(x) 1.0 11 20 21 25 16 0.5 (c). How much money do you expect to draw from your pocket? We expect to draw out E ( X ) = x i x i · p ( x i ) = 11(0 . 1) + 16(0 . 4) + 20(0 . 2) + 21(0 . 1) + 25(0 . 2) = 18 . 6 cents.
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