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

( ) ( ) , if it exists.E Y y f y dy

• Note this parallels our earlier definition for the discrete random variable:

( ) ( )y

E Y y p y

Expected Value, E[g(Y)]

• For a continuous random variable Y, define the expected value of a function of Y as

[ ( )] ( ) ( ) , if it exists.E g Y g y f y dy

• Again, this parallels our earlier definition for the discrete case:

[ ( )] ( ) ( )y

E g Y g y p 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

1 2 1 2[ ( ) ( )] [ ( )] [ ( )]E g Y g Y E g Y E g Y

Properties of Variance

• In the continuous case, our earlier properties for variance also remain valid.

2 2 2( ) [( ) ] ( ) [ ( )]V Y E Y E Y E Y

2( ) ( )V aY b a V Y

and

Problem 4.16

• Find the mean and variance of Y, given

0.2, 1 0( ) 0.2 1.2 , 0 1

0, otherwise

yf Y y y

Problem 4.26• Suppose CPU time used (in hours) has distribution:

2(3/ 64) (4 ), 0 4( )

0, otherwisey y y

f Y

• 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

, for a( )

0, otherwisec y b

f y

• To be a valid density function, it follows that1c

b a

Uniform Distribution

• A continuous random variable has a uniform distribution if its probability density function is given by

1 , for a( )

0, otherwise

y bf y b a

Uniform Mean, Variance

• Upon deriving the expected value and variance for a uniformly distributed random variable, we find

( )2

a bE Y

is the midpoint of the interval

2( )( )12

b aV Y and

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.