1. ELEC 303 – Random Signals Lecture 13 – Transforms Dr. Farinaz Koushanfar ECE Dept., Rice University Oct 15, 2009

2. Lecture outline Reading: 4.4-4.5 Definition and usage of transform Moment generating property Inversion property Examples Sum of independent random variables

3. Transforms The transform for a RV X (a.k.a moment generating function) is a function MX(s) with parameter s defined by M X (s)=E[e sX ]

4. Why transforms? New representation Has usages in – Calculations, e.g., moment generation – Theorem proving – Analytical derivations

5. Example(s) Find the transforms associated with – Poisson random variable – Exponential random variable – Normal random variable – Linear function of a random variable

6. Moment generating property

7. Example Use the moment generation property to find the mean and variance of exponential RV

8. Inverse of transforms Transform M X (s) is invertible – important The transform MX(s) associated with a RV X uniquely determines the CDF of X, assuming that MX(s) is finite for all s in some interval [-a,a], a>0 Explicit formulas that recover PDF/PMF from the associated transform are difficult to use In practice, transforms are often converted by pattern matching

9. Inverse transform – example 1 The transform associated with a RV X is M(s) =1/4 e -s + 1/2 + 1/8 e 4s + 1/8 e 5s We can compare this with the general formula M(s) =  x e sx p X (x) The values of X: -1,0,4,5 The probability of each value is its coefficient PMF:P(X=-1)=1/4; P(X=0)=1/2; P(X=4)=1/8; P(X=5)=1/8;

10. Inverse transform – example 2

11. Mixture of two distributions Example: f X (x)=2/3. 6e -6x + 1/3. 4e -4x,x  0 More generally, let X 1,…,X n be continuous RV with PDFs f X1, f X2,…,f Xn Values of RV Y are generated as follows – Index i is chosen with corresponding prob p i – The value y is taken to be equal to X i f Y (y)= p 1 f X1 (y) + p 2 f X2 (y) + … + p n f Xn (y) M Y (s)= p 1 M X1 (s) + p 2 M X2 (s) + … + p n M Xn (s) The steps in the problem can be reversed then

12. Sum of independent RVs X and Y independent RVs, Z=X+Y M Z (s) = E[e sZ ] = E[e s(X+Y) ] = E[e sX e sY ] = E[e sX ]E[e sY ] = M X (s) M Y (s) Similarly, for Z=X 1 +X 2 +…+X n, M Z (s) = M X1 (s) M X2 (s) … M Xn (s)

13. Sum of independent RVs – example 1 X 1,X 2,…,X n independent Bernouli RVs with parameter p Find the transform of Z=X 1 +X 2 +…+X n

14. Sum of independent RVs – example 2 X and Y independent Poisson RVs with means and  respectively Find the transform for Z=X+Y Distribution of Z?

15. Sum of independent RVs – example 3 X and Y independent normal RVs X ~ N(  x,  x 2 ), and Y ~ N(  y,  y 2 ) Find the transform for Z=X+Y Distribution of Z?

16. Review of transforms so far

17. Bookstore example (1)

18. Sum of random number of independent RVs

19. Bookstore example (2)

20. Transform of random sum

21. Bookstore example (3)

22. More examples… A village with 3 gas station, each open daily with an independent probability 0.5 The amount of gas in each is ~U[0,1000] Characterize the probability law of the total amount of available gas in the village

23. More examples… Let N be Geometric with parameter p Let Xi’s Geometric with common parameter q Find the distribution of Y = X 1 + X 2 + …+ X n