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ELEC 303 – Random SignalsPowerPoint Presentation

ELEC 303 – Random Signals

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### ELEC 303 – Random Signals

Lecture 11 – Derived distributions, covariance, correlation and convolution

Dr. Farinaz Koushanfar

ECE Dept., Rice University

September 29, 2009

Lecture outline

Reading: 4.1-4.2

Derived distributions

Sum of independent random variables

Covariance and correlations

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

Example 1

Let X be uniform on [0,1]

Y=sqrt(X)

FY(y) = P(Yy) = P(Xy2) = y2

fY(y) = dF(y)/dy = d(y2)/dy = 2y 0 y1

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

The linear case

- If Y=aX+b, for a and b scalars and a0
- Example 1: Linear transform of an exponential RV (X): Y=aX+b
- fX(x) = e-x, for x0, and otherwise fX(x)=0

- Example 2: Linear transform of normal RV

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:

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?

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?

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)

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]

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