MC ontinuousMoments

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					Mean and Variance for
 Continuous R.V.s
         Expected Value, E(Y)
• For a continuous random variable Y, define the
  expected value of Y as
                    
          E (Y )   y f ( y)dy, if it exists.
                    


• Note this parallels our earlier definition for the
  discrete random variable:
                E (Y )   y p ( y )
                          y
       Expected Value, E[g(Y)]
• For a continuous random variable Y, define the
  expected value of a function of Y as
                     
      E[ g (Y )]   g ( y) f ( y)dy, if it exists.
                    


• Again, this parallels our earlier definition for the
  discrete case:
             E[ g (Y )]   g ( y ) p( y )
                           y
   Properties of Expected Value
• In the continuous case, all of our earlier properties
  for working with expected value are still valid.
                
      E (c)   c f ( y)dy  c
                

      E (aY  b)  aE (Y )  b
      E[ g1 (Y )  g 2 (Y )]  E[ g1 (Y )]  E[ g 2 (Y )]
         Properties of Variance
• In the continuous case, our earlier properties for
  variance also remain valid.

    V (Y )  E[(Y   )2 ]  E(Y 2 )  [ E(Y )]2
   and

    V (aY  b)  a V (Y )
                     2
               Problem 4.16
• Find the mean and variance of Y, given

                 0.2,         1  y  0
                
       f (Y )   0.2  1.2 y, 0  y  1
                 0,
                             otherwise
                 Problem 4.26
• Suppose CPU time used (in hours) has
  distribution:
                (3/ 64) y 2 (4  y), 0  y  4
       f (Y )  
                0,                   otherwise

• Find the mean and variance of Y.
• If CPU charges are $200 per hour, find the mean
  and variance for CPU charges.
• Do you expect the CPU charge to exceed $600
  very often?
The Uniform Distribution
                Equally Likely
• If Y takes on values in an interval (a, b) such that
  any of these values is equally likely, then

           c, for a  y  b
  f ( y)  
            0, otherwise

• To be a valid density function, it follows that
                         1
                     c
                        ba
          Uniform Distribution
• A continuous random variable has a uniform
  distribution if its probability density function is
  given by

                      1
                            , for a  y  b
            f ( y)   b  a
                      0, otherwise
                     
      Uniform Mean, Variance
• Upon deriving the expected value and variance for
  a uniformly distributed random variable, we find

                          ab
                 E (Y ) 
                           2
          is the midpoint of the interval

                       (b  a)2
          and V (Y ) 
                          12
                    Example
• Suppose the round-trip times for deliveries from a
  store to a particular site are uniformly distributed
  over the interval 30 to 45 minutes.
• Find the probability the delivery time exceeds 40
  minutes.
• Find the probability the delivery time exceeds 40
  minutes, given it exceeds 35 minutes.
• Determine the mean and variance for these
  delivery times.
              Problem 4.42
• In an experiment, times are recorded and
  the measurement errors are assumed to be
  uniformly distributed between
  – 0.05 and + 0.05 s (“microseconds”).
• Find the probability the measurement is
  accurate to within 0.01 s.
• Find the mean and variance for the errors.

				
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posted:10/5/2012
language:English
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