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ELEC 303 – Random Signals

ELEC 303 – Random Signals. Lecture 13 – Transforms Dr. Farinaz Koushanfar ECE Dept., Rice University Oct 15, 2009. Lecture outline. Reading: 4.4-4.5 Definition and usage of transform Moment generating property Inversion property Examples Sum of independent random variables. Transforms.

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ELEC 303 – Random Signals

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  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.amoment generating function) is a function MX(s) with parameter s defined by MX(s)=E[esX]

  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 MX(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 e4s + 1/8 e5s • We can compare this with the general formula • M(s) = xesxpX(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: fX(x)=2/3. 6e-6x + 1/3. 4e-4x, x0 • More generally, let X1,…,Xn be continuous RV with PDFs fX1, fX2,…,fXn • Values of RV Y are generated as follows • Index i is chosen with corresponding prob pi • The value y is taken to be equal to Xi fY(y)= p1 fX1(y) + p2 fX2(y) + … + pnfXn(y) MY(s)= p1 MX1(s) + p2 MX2(s) + … + pnMXn(s) • The steps in the problem can be reversed then

  12. Sum of independent RVs • X and Y independent RVs, Z=X+Y • MZ(s) = E[esZ] = E[es(X+Y)] = E[esXesY] = E[esX]E[esY] = MX(s) MY(s) • Similarly, for Z=X1+X2+…+Xn, MZ(s) = MX1(s) MX2(s) … MXn(s)

  13. Sum of independent RVs – example 1 • X1,X2,…,Xnindependent Bernouli RVs with parameter p • Find the transform of Z=X1+X2+…+Xn

  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,x2), and Y ~ N(y,y2) • 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 = X1+ X2+ …+ Xn

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