1ME 233 Advanced Control II

Lecture 4
Introduction to Probability TheoryRandom Vectors and Conditional Expectation(ME233 Class Notes pp. PR4-PR6)

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Outline• Multiple random variables• Random vectors– Correlation and covariance• Gaussian random variables• PDFs of Gaussian random vectors• Conditional expectation of Gaussian randomvectors

3Multiple Random Variables
Let X and Y be continuous random variables.

• Their joint cumulative distribution function(CDF) is given by

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Multiple Random Variables
Let X and Y be continuous random variables
with a differentiable joint CDF

Their joint probability density function (PDF) is5Multiple Random Variablesp (x,y)XYpXY(a,c)p (b,c)caxb

XY

pXY(b,d)XY

has the usual
meaning of densityp (a,d)dy

6Multiple Random Variables
Let X and Y be independent• Then:Marginal CDF of X

Marginal CDF of Y

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Multiple Random Variables
Let X and Y be independent• Then:Marginal PDF of X

Marginal PDF of Y

8Correlation and Covariance
Let X and Y be continuous random variables
with joint PDF• Correlation:

9Mean
Let X and Y be continuous random variables
with joint PDF• Mean:

where

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Correlation and Covariance
Let X and Y be continuous random variables
with joint PDF• Covariance:

means

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Correlation and Covariance
Let X and Y be continuous random variables
with joint PDF• X and Y are uncorrelated if :their covariance is zero

•X and Y are orthogonal if :their correlation is zero

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Multiple Random Variables• X and Y are uncorrelated if and only if

Proof:

therefore13Variance
The variance of random variable X is:14Marginal PDF
Let X and Y have a joint PDF• Marginal or unconditional PDFs:

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Marginal PDF
Let X and Y have a joint PDF• Expected value of X

16Conditional PDF
Let X and Y have a joint PDF• The Conditional PDF of X given an
outcome of Y = y1 :

17Conditional PDF
Let X and Y have a joint PDF• The Conditional PDF of Y given an
outcome of X = x1 :

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Conditional PDF
Let X and Y have a joint PDF• Bayes’ rule:19Conditional Expectation
Let X and Y have a joint PDF• Conditional Expectation of X given anoutcome of Y = y1 :

20Conditional Variance
Let X and Y have a joint PDF• Conditional variance of X given an outcomeof Y = y1 :

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Independent Variables
Let X and Y be independent. Then:

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Independent Variables
If X and Y are independent random variables,
then X and Y are uncorrelated
Proof:(independence)

The converse statement is NOT true in general

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Bilateral Laplace and Fourier Transforms
Given• Laplace transform:• Inverse Laplace transform:

for some real γ so that contour path of integration
is in the region of convergence24Bilateral Laplace and Fourier Transforms
Given• Fourier transform:• Inverse Fourier transform:

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Moment Generating Function
The Fourier transform of the PDF of a random variable
X is also called the moment generating function or
characteristic function
Notice that, given the PDF pX(x)it can be shown that
where [n] indicates the nth derivative w/r ω (see Poolla’s notes)

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Properties of Normal distributions
The moment generating function of a zeromean normal distribution is also normal.

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Moment generating functions of Normal PDFs
Let,i.e.,

The moment generating functions of X is:

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Sum of independent random variables
Let X and Y be two independent random variables
with PDFs
Definethen(convolution)

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Proof
Assume X and Y are two independent randomvariables and define
Let us now calculate the moment generating
function of Z:(independence)

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ProofSince

Applying the inverse Fourier transform,

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Random Vectors
Let X1 and X2 be continuous random variables.
Recall that:• Their joint CDF is given by• Their joint PDF is32

Random Vector
Define the random vector(and the dummy vector)

with CDF

33Random Vector
Define the random vector(and the dummy vector)

with PDF

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Random Vector
Define the random vector
Mean:

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Random Vector
Define the random vector
Mean:

MarginalPDFs

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Correlation

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Covariance38

Covariance

Proof:• Define any deterministic vector•

is a scalar random variable.

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Random Vectors
X be a random n vector

Y be a random m vector

with PDF

with PDF

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Cross-covariance
X be a random n vector

Y be a random m vector

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Cauchy-Schwarz inequality
For any scalar random variables X and Y42

Proof
Define the random vectorThus,

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Gaussian Random Variables (Review)
Let X be Gaussian with PDF

Frequently-used notation
X is normally distributed withmeanand variance44

Two independent Gaussians

0.14

0.2
0.18

0.12

0.16

0.1

0.14
0.12

0.08

0.1

0.06
0.08
0.06

0.04

0.04

0.02
0.02
0-10-505

100-10-5051045

Space-saving notation

dummy variables

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Two independent Gaussians

0.03
0.025

0.03

0.02

-100.015-5-100.025
0.02-50.015

0.01
00.00550-10-8-6-40.01010-2024681000.005
856420-2-4-6-8-101047

Two independent Gaussians
Joint PDF of independent Gaussian X and Y

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Two independent Gaussians
Joint PDF of independent Gaussian X and Y

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Two independent Gaussians
Define the vector(independent Gaussian X and Y)

Covariance50

Two independent Gaussians
Joint PDF of independent Gaussian X and Y

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Two independent Gaussians
Joint PDF of independent Gaussian X and Y

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2-dimensional Gaussian random vector
X and Yindependent53n-dimensional Gaussian random vector
Joint PDF of a Gaussian vector

n: dimension of Z

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Linear combination of Gaussians
If X is Gaussian andZ = AX + bwhere• A is a deterministic matrix• b is a deterministic vector
then Z is also Gaussian

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Conditional PDF (Review)
Let X and Y have a joint PDF• The Conditional PDF of X given an
outcome of Y = y1 :

56Conditional Expectation (Review)
Let X and Y have a joint PDF• Conditional Expectation of X given an
outcome of Y = y1 :

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Motivation for Gaussians
When X and Y are Gaussians
The conditional probabilities

and conditional expectations(for any outcome y )can be calculated very easily!

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Random Vectors
Define the Gaussian random n + m vector

X is Gaussian n vector Y is a Gaussian m vector

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Random Vectors
X is Gaussian n vector

Y is a Gaussian m vector(n × n)(m × m)(n × m)60

Conditional PDF for Gaussians• The conditional PDF of X given Y = yalso a Gaussian PDF

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Conditional PDF for Gaussians
The conditional random vector X given andoutcome Y = yis also normally distributed(also a Gaussian random vector)

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Conditional PDF for Gaussiansconditional expectation of X given Y = yaffine function of the outcome y63Conditional PDF for Gaussians

The conditional covariance of X given Y = y
independent of the outcome y !!

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Conditional covariance of X given Y = y

max eigenvalues

min eigenvalue

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Independent Gaussians
Let X and Y be jointly Gaussian random vectors.
X and Y are independent if and only if they are uncorrelated
Proof:We already showed this this is true even if X and Y are
not jointly Gaussian

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Proof of conditional PDF for Gaussians
Idea of proof• Some details regarding Schur complements• A lot of algebra…

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Schur complement• Given• Then• Schur complement of B:

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Schur complement• Given• If Schur complement of B

is nonsingular• Then

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Proof• Given• Define• Then• Results follow by computing inverses anddeterminants of matrices Q and R

70details

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Conditional covariance• Given• The Schur complement of

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Schur complement of• Given• Then

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Schur complement of• Given• and74

TheoremGiven

Thenwith

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Proof
• Random vector•dummy variables

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Proof: use Schur complement• Now compute:• Using:

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Proof
• Now compute:

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Proof: compute the conditional PDF

where:dimension of Y

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Proof: compute the conditional PDF

where:

dimension of X + dimension of Y

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Proof

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Proof

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Proof

use Schur determinant result:

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Proof

Now use:

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Proof

Now use:

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Proof

Therefore,

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Proof
Therefore,

withThis result is important and constitutes thebasis for the Kalman Filter!

Supplemental Material(You are not responsible for this…)• Laplace and Fourier transform of Gaussian
PDF• Transformation of random variables

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Laplace transform of normal PDF

where, after “completing the squares”,

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Laplace transform of normal PDF
substituting,= 1 (area under a PDF = 1)

Fourier transform:

90Transformation of random variables
Given a real valued function f of random variable X

Assume that Y is also a random variable.
Also assume thatexists. Then,

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Transformation of random variables
Letand