1. ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 11 – Derived distributions, covariance, correlation and convolution Dr. Farinaz Koushanfar ECE Dept., Rice University September 29, 2009

2. ELEC 303, Koushanfar, Fall’09 Lecture outline Reading: 4.1-4.2 Derived distributions Sum of independent random variables Covariance and correlations

3. ELEC 303, Koushanfar, Fall’09 Derived distributions Consider the function Y=g(X) of a continuous RV X Given PDF of X, we want to compute the PDF of Y The method – Calculate CDF FY(y) by the formula – Differentiate to find PDF of Y

4. ELEC 303, Koushanfar, Fall’09 Example 1 Let X be uniform on [0,1] Y=sqrt(X) FY(y) = P(Y  y) = P(X  y 2 ) = y 2 f Y (y) = dF(y)/dy = d(y 2 )/dy = 2y0  y  1

5. ELEC 303, Koushanfar, Fall’09 Example 2 John is driving a distance of 180 miles with a constant speed, whose value is ~U[30,60] miles/hr Find the PDF of the trip duration? Plot the PDF and CDFs

6. ELEC 303, Koushanfar, Fall’09 Example 3 Y=g(X)=X 2, where X is a RV with known PDF Find the CDF and PDF of Y?

7. ELEC 303, Koushanfar, Fall’09 The linear case If Y=aX+b, for a and b scalars and a  0 Example 1: Linear transform of an exponential RV (X): Y=aX+b – f X (x) = e - x, for x  0, and otherwise f X (x)=0 Example 2: Linear transform of normal RV

8. ELEC 303, Koushanfar, Fall’09 The strictly monotonic case X is a continuous RV and its range in contained in an interval I Assume that g is a strictly monotonic function in the interval I Thus, g can be inverted: Y=g(X) iff X=h(Y) Assume that h is differentiable The PDF of Y in the region where f Y (y)>0 is:

9. ELEC 303, Koushanfar, Fall’09 More on strictly monotonic case

10. ELEC 303, Koushanfar, Fall’09 Example 4 Two archers shoot at a target The distance of each shot is ~U[0,1], independent of the other shots What is the PDF for the distance of the losing shot from the center?

11. ELEC 303, Koushanfar, Fall’09 Example 5 Let X and Y be independent RVs that are uniformly distributed on the interval [0,1] Find the PDF of the RV Z?

12. ELEC 303, Koushanfar, Fall’09 Sum of independent RVs - convolution

13. ELEC 303, Koushanfar, Fall’09 X+Y: Independent integer valued

14. ELEC 303, Koushanfar, Fall’09 X+Y: Independent continuous

15. ELEC 303, Koushanfar, Fall’09 X+Y Example: Independent Uniform

16. ELEC 303, Koushanfar, Fall’09 X+Y Example: Independent Uniform

17. ELEC 303, Koushanfar, Fall’09 Two independent normal RVs

18. ELEC 303, Koushanfar, Fall’09 Sum of two independent normal RVs

19. ELEC 303, Koushanfar, Fall’09 Covariance Covariance of two RVs is defined as follows An alternate formula: Cov(X,Y) = E[XY] – E[X]E[Y] Properties – Cov(X,X) = Var(X) – Cov(X,aY+b) = a Cov(X,Y) – Cov(X,Y+Z) = Cov(X,Y) + Cov (Y,Z)

20. ELEC 303, Koushanfar, Fall’09 Covariance and correlation If X and Y are independent  E[XY]=E[X]E[Y] So, the cov(X,Y)=0 The converse is not generally true!! The correlation coefficient of two RVs is defined as The range of values is between [-1,1]

21. ELEC 303, Koushanfar, Fall’09 Variance of the sum of RVs Two RVs: Multiple RVs