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
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Lecture 11 – Derived distributions, covariance, correlation and convolution
Dr. Farinaz Koushanfar
ECE Dept., Rice University
September 29, 2009
Sum of independent random variables
Covariance and correlations
Let X be uniform on [0,1]
FY(y) = P(Yy) = P(Xy2) = y2
fY(y) = dF(y)/dy = d(y2)/dy = 2y 0 y1
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
Y=g(X)=X2, where X is a RV with known PDF
Find the CDF and PDF of Y?
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:
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?
Let X and Y be independent RVs that are uniformly distributed on the interval [0,1]
Find the PDF of the RV Z?
Cov(X,Y) = E[XY] – E[X]E[Y]
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]