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

Expected Value, E[g(Y)] For a continuous random variable Y, define the expected value of a function of Y as Again, this parallels our earlier definition for the discrete case:

Properties of Expected Value In the continuous case, all of our earlier properties for working with expected value are still valid.

Properties of Variance In the continuous case, our earlier properties for variance also remain valid. and

Problem 4.16 Find the mean and variance of Y, given

Problem 4.26 Suppose CPU time used (in hours) has distribution: 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 To be a valid density function, it follows that

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

Uniform Mean, Variance Upon deriving the expected value and variance for a uniformly distributed random variable, we find is the midpoint of the interval 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  s (“microseconds”). Find the probability the measurement is accurate to within 0.01  s. Find the mean and variance for the errors.