ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 9 – Continuous Random Variables: Joint PDFs, Conditioning, Continuous Bayes Farinaz Koushanfar ECE Dept., Rice University Sept 23, 2009

ELEC 303, Koushanfar, Fall’09 Lecture outline Reading: Reading Continuous RV, PDF, CDF (review) Joint PDF and multiple RVs Conditioning Independence

ELEC 303, Koushanfar, Fall’09 PDF (review) A RV is continuous if there is a non-negative PDF s.t. for every subset B of real numbers: The probability that RV X falls in an interval is: Figure courtesy of Bertsekas&Tsitsiklis, Introduction to Probability, 2008

ELEC 303, Koushanfar, Fall’09 PDF (Cont’d) Continuous prob – area under the PDF graph For any single point: The PDF function (f X ) non-negative for every x Area under the PDF curve should sum up to 1

ELEC 303, Koushanfar, Fall’09 Mean and variance (review) Expectation E[X] and n-th moment E[X n ] are defined similar to discrete A real-valued function Y=g(X) of a continuous RV is a RV: Y can be both continous or discrete

ELEC 303, Koushanfar, Fall’09 Properties of CDF (review) Defined by: F X (x) = P(X  x), for all x F X (x) is monotonically nondecreasing – If x<y, then F X (x)  F X (y) – F X (x) tends to 0 as x  - , and tends to 1 as x   – For discrete X, F X (x) is piecewise constant – For continuous X, F X (x) is a continuous function – PMF and PDF obtained by summing/differentiate

ELEC 303, Koushanfar, Fall’09 Standard normal RV (review) A continuous RV is standard normal or Gaussian N(0,1), if

ELEC 303, Koushanfar, Fall’09 Notes about normal RV (review) Normality preserved under linear transform It is symmetric around the mean No closed form is available for CDF Standard tables available for N(0,1), E.g., p155 The usual practice is to transform to N(0,1): – Standardize X: subtract  and divide by  to get a standard normal variable y – Read the CDF from the standard normal table

ELEC 303, Koushanfar, Fall’09 Joint PDFs of multiple RV Joint PDF fX,Y, where this is a nonnegative function Interpretation

ELEC 303, Koushanfar, Fall’09 Marginal PDFs Consider the event x  A Compare with the formula Thus, the marginal PDF f X is given by

ELEC 303, Koushanfar, Fall’09 Two-dimensional Uniform PDF Compute c?

ELEC 303, Koushanfar, Fall’09 Buffon’s needle (2)

ELEC 303, Koushanfar, Fall’09 Buffon’s needle (2)

ELEC 303, Koushanfar, Fall’09 Buffon’s needle (3)

ELEC 303, Koushanfar, Fall’09 Buffon’s needle (4)

ELEC 303, Koushanfar, Fall’09 Joint CDF, expectation F X,Y (x,y) = PDF can be found from CDF by differentiating: Expectation Expectation is additive and linear

ELEC 303, Koushanfar, Fall’09 More than two RVs The joint PDF for more RVs is similar Marginal Expectation of sum

ELEC 303, Koushanfar, Fall’09 Conditioning A RV on an event If we condition on an event of form X  A, with P(X  A)>0, then we have By comparing, we get

ELEC 303, Koushanfar, Fall’09 Example: the exponential RV The time t until a light bulb dies is an exponential RV with parameter If one turns the light, leaves the room and return t seconds later(A={T>t}) X is the additional time until bulb is burned What is the conditional CDF of X, given A? Memoryless property of exponential CDF!

ELEC 303, Koushanfar, Fall’09 Example: total probability theorem Train arrives every 15 mins startng 6am You walk to the station between 7:10-7:30am Your arrival is uniform random variable Find the PDF of the time you have to wait for the first train to arrive x f X (x) y f y|A (y) 7:107:30 5 1/5 y f y|B (y) 15 1/15 y f y (y) 5 1/ /20

ELEC 303, Koushanfar, Fall’09 Conditioning a RV on another The conditional PDF Can use marginal to compute f Y (y) Note that we have

ELEC 303, Koushanfar, Fall’09 Summary of concepts Courtesy of Prof. Dahleh, MIT

ELEC 303, Koushanfar, Fall’09 Conditional expectation Definitions The expected value rule Total expectation theorem

ELEC 303, Koushanfar, Fall’09 Mean and variance of a piecewise constant PDF Consider the events – A 1 ={x is in the first interval [0,1] – A 2 ={x is in the second interval (1,2] Find P(A 1 ), P(A 2 )? Use total expectation theorem to find E[X] and Var(X)?

ELEC 303, Koushanfar, Fall’09 Example: stick breaking (1)

ELEC 303, Koushanfar, Fall’09 Example: stick breaking (2)

ELEC 303, Koushanfar, Fall’09 Example: stick breaking (3)

ELEC 303, Koushanfar, Fall’09 Independence Two RVs X and Y are independent if This is the same as (for all y with f Y (y)>0) Can be easily generalized to multiple RVs