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

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ELEC 303 – Random Signals

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Elec 303 random signals

ELEC 303 – Random Signals

Lecture 11 – Derived distributions, covariance, correlation and convolution

Dr. Farinaz Koushanfar

ECE Dept., Rice University

September 29, 2009


Lecture outline

Lecture outline

Reading: 4.1-4.2

Derived distributions

Sum of independent random variables

Covariance and correlations


Derived distributions

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

Example 1

Let X be uniform on [0,1]

Y=sqrt(X)

FY(y) = P(Yy) = P(Xy2) = y2

fY(y) = dF(y)/dy = d(y2)/dy = 2y0 y1


Example 2

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


Example 3

Example 3

Y=g(X)=X2, where X is a RV with known PDF

Find the CDF and PDF of Y?


The linear case

The linear case

  • If Y=aX+b, for a and b scalars and a0

  • Example 1: Linear transform of an exponential RV (X): Y=aX+b

    • fX(x) = e-x, for x0, and otherwise fX(x)=0

  • Example 2: Linear transform of normal RV


The strictly monotonic case

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:


More on strictly monotonic case

More on strictly monotonic case


Example 4

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

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?


Sum of independent rvs convolution

Sum of independent RVs - convolution


X y independent integer valued

X+Y: Independent integer valued


X y independent continuous

X+Y: Independent continuous


X y example independent uniform

X+Y Example: Independent Uniform


X y example independent uniform1

X+Y Example: Independent Uniform


Two independent normal rvs

Two independent normal RVs


Sum of two independent normal rvs

Sum of two independent normal RVs


Covariance

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

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]


Variance of the sum of rvs

Variance of the sum of RVs

Two RVs:

Multiple RVs


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