ELEC 303, Koushanfar, Fall’09

ELEC 303 – Random Signals

Lecture 11 – Derived distributions, covariance, correlation and convolution

Dr. Farinaz KoushanfarECE Dept., Rice University

September 29, 2009

ELEC 303, Koushanfar, Fall’09

Lecture outline

• Reading: 4.1-4.2• Derived distributions• Sum of independent random variables• Covariance and correlations

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

ELEC 303, Koushanfar, Fall’09

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

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

ELEC 303, Koushanfar, Fall’09

Example 3

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

ELEC 303, Koushanfar, Fall’09

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

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 fY(y)>0 is:

)())(()( ydy

dhyhfyf XY

ELEC 303, Koushanfar, Fall’09

More on strictly monotonic case

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?

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?

ELEC 303, Koushanfar, Fall’09

Sum of independent RVs - convolution

ELEC 303, Koushanfar, Fall’09

X+Y: Independent integer valued

ELEC 303, Koushanfar, Fall’09

X+Y: Independent continuous

ELEC 303, Koushanfar, Fall’09

X+Y Example: Independent Uniform

ELEC 303, Koushanfar, Fall’09

X+Y Example: Independent Uniform

ELEC 303, Koushanfar, Fall’09

Two independent normal RVs

ELEC 303, Koushanfar, Fall’09

Sum of two independent normal RVs

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)

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]

ELEC 303, Koushanfar, Fall’09

Variance of the sum of RVs

• Two RVs:

• Multiple RVs