Understanding Continuous Random Variables: PDFs and CDFs in Probability Theory

1. ELEC 303 – Random Signals Lecture 8 – Continuous Random Variables: PDF and CDFs FarinazKoushanfar ECE Dept., Rice University Sept 18, 2009

2. Lecture outline Reading: Reading 3.1-3.3 Continuous random variables Probability density function (PDF) Cumulative density function (CDF) Normal random variable

3. 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

4. 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

5. PDF (Cont’d) Continuous prob – area under the PDF graph For any single point: The PDF function (fX) non-negative for every x Area under the PDF curve should sum up to 1

6. PDF (example) A PDF can take arbitrary value, as long as it is summed to one over the interval, e.g.,

7. Mean and variance Expectation E[X] and n-th moment E[Xn] 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

8. Mean and variance of Uniform RV

9. Exponential RV fX(x)  • Mean? • Variance?  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

10. Cumulative distribution function (CDF) The CDF of a RV X is denoted by FX and provides the probability P(Xx). For every x, Uniform example: Defined for both continuous and discrete RVs

11. CDF of discrete RV

12. Properties of CDF • Defined by: FX(x) = P(Xx), for all x • FX(x) is monotonically nondecreasing • If x<y, then FX(x)  FX(y) • FX(x) tends to 0 as x-, and tends to 1 as x • For discrete X, FX(x) is piecewise constant • For continuous X, FX(x) is a continuous function • PMF and PDF obtained by summing/differentiate

13. Example You are allowed to take an exam 3 times and final score is the max of 3: X=max(X1,X2,X3) Scores are independent uniform from [1,10] What is the PMF? pX(k)=FX(k)-FX(k-1), k=1,…,10 FX(k)=P(Xk) = P(X1k, X2k, X3k) = P(X1k) P(X2k) P(X3k) = (k/10)3 PX(k)=(K/10)3 – ((k-1)/10)3

14. Geometric and exponential CDFs n=1,2,… 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

15. Standard Gaussian (normal) RV A continuous RV is standard normal or Gaussian N(0,1), if

16. General Gaussian RV

17. 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