# 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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 views: 2 posted: 10/5/2012 language: English pages: 13