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

2. Lecture outline Reading: 4.1-4.2 Derived distributions Sum of independent random variables Covariance and correlations

3. 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. Example 1 Let X be uniform on [0,1] Y=sqrt(X) FY(y) = P(Yy) = P(Xy2) = y2 fY(y) = dF(y)/dy = d(y2)/dy = 2y 0 y1

5. 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. Example 3 Y=g(X)=X2, where X is a RV with known PDF Find the CDF and PDF of Y?

7. 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 • fX(x) = e-x, for x0, and otherwise fX(x)=0 • Example 2: Linear transform of normal RV

8. 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 fY(y)>0 is:

9. More on strictly monotonic case

10. 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. 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. Sum of independent RVs - convolution

13. X+Y: Independent integer valued

14. X+Y: Independent continuous

15. X+Y Example: Independent Uniform

16. X+Y Example: Independent Uniform

17. Two independent normal RVs

18. Sum of two independent normal RVs

19. 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. 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. Variance of the sum of RVs Two RVs: Multiple RVs