# hw3-sol - AMS 311(Fall 2009 Joe Mitchell PROBABILITY THEORY...

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AMS 311 (Fall, 2009) Joe Mitchell PROBABILITY THEORY Homework Set # 3 – Solution Notes (1). (30 points) A box contains 10 dimes, 15 nickels, and 4 pennies. We draw 2 coins at random (without replacement). Let X represent the amount (in cents) that we draw out of the box. (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 { 2 , 6 , 10 , 11 , 15 , 20 } . The probability function is given by p ( x ) = P ( X = x ) = ( 4 2 ) ( 29 2 ) if x = 2 ( 4 1 )( 15 1 ) ( 29 2 ) if x = 6 ( 15 2 ) ( 29 2 ) if x = 10 ( 4 1 )( 10 1 ) ( 29 2 ) if x = 11 ( 15 1 )( 10 1 ) ( 29 2 ) if x = 15 ( 10 2 ) ( 29 2 ) if x = 20 0 otherwise The plot of p ( x ) simply consists of “spikes” of the appropriate heights at x = 2 , 6 , 10 , 11 , 15 , 20, with p ( x ) = 0 everywhere else. (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 < 2 ( 4 2 ) ( 29 2 ) if 2 x < 6 ( 4 2 ) ( 29 2 ) + ( 4 1 )( 15 1 ) ( 29 2 ) if 6 x < 10 ( 4 2 ) ( 29 2 ) + ( 4 1 )( 15 1 ) ( 29 2 ) + ( 15 2 ) ( 29 2 ) if 10 x < 11 ( 4 2 ) ( 29 2 ) + ( 4 1 )( 15 1 ) ( 29 2 ) + ( 15 2 ) ( 29 2 ) + ( 4 1 )( 10 1 ) ( 29 2 ) if 11 x < 15 ( 4 2 ) ( 29 2 ) + ( 4 1 )( 15 1 ) ( 29 2 ) + ( 15 2 ) ( 29 2 ) + ( 4 1 )( 10 1 ) ( 29 2 ) + ( 15 1 )( 10 1 ) ( 29 2 ) if 15 x < 20 1 if x 20 (c). How much money do we expect to draw from the box?

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