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 - PowerPoint PPT PresentationTRANSCRIPT
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:
)())(()( ydydhyhfyf 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