ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 8 – Continuous Random Variables: PDF and CDFs Farinaz Koushanfar ECE Dept., Rice University Sept 18, 2009

ELEC 303, Koushanfar, Fall’09 Lecture outline Reading: Reading Continuous random variables Probability density function (PDF) Cumulative density function (CDF) Normal random variable

ELEC 303, Koushanfar, Fall’09 Continuous random variables Random variables with a continuous range of values – E.g., speedometer, people’s height, weight Possible to approximate with discrete Continuous models are useful – Fine-grain and more accurate – Continuous calculus tools – More insight from analysis

ELEC 303, Koushanfar, Fall’09 Probability density functions (PDFs) 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 PDF (example) A PDF can take arbitrary value, as long as it is summed to one over the interval, e.g.,

ELEC 303, Koushanfar, Fall’09 Mean and variance 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 Mean and variance of Uniform RV

ELEC 303, Koushanfar, Fall’09 Exponential RV is a positive RV characterizing the PDF E.g., time interval between two packet arrivals at a router, the lift time of a bulb The probability that X exceeds a certain value decreases exponentially, for any (a  0) we have f X (x) Mean? Variance?

ELEC 303, Koushanfar, Fall’09 Cumulative distribution function (CDF) The CDF of a RV X is denoted by F X and provides the probability P(X  x). For every x, Uniform example: Defined for both continuous and discrete RVs

ELEC 303, Koushanfar, Fall’09 CDF of discrete RV

ELEC 303, Koushanfar, Fall’09 Properties of CDF 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 Example You are allowed to take an exam 3 times and final score is the max of 3: X=max(X 1,X 2,X 3 ) Scores are independent uniform from [1,10] What is the PMF? p X (k)=F X (k)-F X (k-1),k=1,…,10 F X (k)=P(X  k) = P(X 1  k, X 2  k, X 3  k) = P(X 1  k) P(X 2  k) P(X 3  k) = (k/10) 3 P X (k)=(K/10) 3 – ((k-1)/10) 3

ELEC 303, Koushanfar, Fall’09 Geometric and exponential CDFs CDF of a Geometric RV with parameter p (A is number of trials before the first success): For an exponential RV with parameter >0, The exponential RVs can be interpreted as the limit for the Geometric RV n=1,2,…

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

ELEC 303, Koushanfar, Fall’09 General Gaussian RV

ELEC 303, Koushanfar, Fall’09 Notes about normal RV 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