Document Sample

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 ba 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 ab 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.

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 2 |

posted: | 10/5/2012 |

language: | English |

pages: | 13 |

OTHER DOCS BY X9RE37qi

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.