ece 6540, lecture 01

Variations

ECE 6540, Lecture 01Introduction and Review of Probability

Estimation Theory: Some Definitions

Definitions Question: What is a statistic? How do we define it?

Answer: A statistic is any function of sampled data

The function must be independent of the data’s underlying probability distribution

Definitions Examples:

𝑥𝑥~𝒩𝒩 0,1 , y~𝒩𝒩 0,2

Are these statistics? 𝑥𝑥

𝑥𝑥 + 𝑦𝑦

𝑥𝑥2

𝑥𝑥𝑥𝑥 − 𝑦𝑦ln 𝑥𝑥 + 2 + 3𝑦𝑦

𝐸𝐸 𝑥𝑥

𝑥𝑥~𝒩𝒩 0,1 , y~𝒩𝒩 0,2

Are these statistics? 𝑥𝑥 Yes!

𝑥𝑥 + 𝑦𝑦 Yes!

𝑥𝑥2 Yes!

𝑥𝑥𝑥𝑥 − 𝑦𝑦ln 𝑥𝑥 + 2 + 3𝑦𝑦 Yes!

𝐸𝐸 𝑥𝑥 No!

Definitions Question: What is an estimator? How do we define it?

Answer: An estimator is a statistic that estimates a specific value

𝑥𝑥~𝒩𝒩 0,1 , y~𝒩𝒩 0,1

A familiar statistic12𝑥𝑥 + 𝑦𝑦 is an estimator of what?

Is it a good estimator? Why or why not?

𝑥𝑥~𝒩𝒩 0,1 , y~𝒩𝒩 0,1

A less familiar statistic23𝑥𝑥 + 1

3𝑦𝑦 is an estimator of what?

Is it a good estimator? Why or why not?

Definitions Question: What is an estimation theory? How do we define it?

Answer: Estimation theory is the study estimators and their properties.

Definitions Question: What is a [statistical] detector (not to be confused with communications

detector)?

Definitions Question: What is a [statistical] detector (not to be confused with communications

detector)?

Answer: (Warning: definition is a but fuzzy) A detector is a statistic or process

that determines the presence of a signal within noise

A [hypothesis] test is a method for determining what distribution a detector (statistic) belongs to

Definitions Example:

𝑛𝑛~𝒩𝒩 0,1 , 𝑥𝑥 is any signal

Hypothesis Test Null Hypothesis: y = n

Alternative Hypothesis: y = x + n

What is a good detector?

Definitions Example:

𝑛𝑛~𝒩𝒩 0,1 , 𝑥𝑥 is any signal

Hypothesis Test Null Hypothesis: y = n

Alternative Hypothesis: y = x + n

What is a good detector? Optimal detector: s = 𝑦𝑦 2

Optimal test: s > 𝜆𝜆 (threshold 𝜆𝜆 is determined from a Chi-square distribution)

Definitions Question: What is an detection theory? How do we define it?

Answer: Detection theory is the study detectors and their properties.

Definitions Question: In engineering, how do we define the “best” or “optimal” of something

(e.g., the best estimator or the best detector)

Definitions Question: In engineering, how do we define the “best” or “optimal” of something

(e.g., the best estimator or the best detector)

Answer: Trick question, “best” and “optimal” is always based on some criteria

that WE define.

Applications

Definitions Question: What are some applications of estimation theory?

Applications RADAR (Radio Detection And Ranging) / Sonar

Detection: Is there a reflection from an aircraft?

Estimation: How far is the aircraft / what is its precise location?

Related (Waveform Design): Can I design waveforms to make the above easier/harder?

Detection Theory Estimation Theory

Credit: https://en.wikipedia.org/wiki/Radar23

Applications Communications Detection: Did I receive a message?

Estimation: What is the message?

Related (Coding): Can I design codes to make the above easier/harder?

Credit: http://www.ohlone.edu/instr/speech/longdesc-diagramcommunication.html

Applications It is pervasive in signal processing Estimation theory = applied statistics

Most modern signal processing tools involve statistics— Array processing— Compressive sensing (most proofs are probability based)— Network science (probabilistic graphical models)— Optimal filter design— De-noising — Tracking (e.g., Kalman filter)— Statistical Modelling / Analysis

Applications Two examples: Gaussian Random Variable

— What is an optimal estimate for the expected value?

Laplace Random Variable— What is an optimal estimate for the expected value?

Schedule

Schedule Part 1: Classical Estimation Theory Minimum Mean Square Error Estimators

Minimum Variance Unbiased Estimators

Maximum Likelihood Estimators

Part 2: Bayesian Estimation Theory Maximum a Priori Estimators

Minimax Estimators

Part 3: Detection Theory Neyman-Pearson Tests

Generalized Maximum Likelihood Tests

Schedule Question: What is the difference between classical and Bayesian statistics? Why are these differences important?

Schedule Quick warning! In the first few classes, I am going to throw a lot of information at you

(some of which you may know, some of which you may not)

I do not expect you to retain everything 100%. My goal is to expose you to these concepts and make you more comfortable about the concepts.

Probability Review (with some things you may have not seen)

Probability Review Quick Note: Notation everywhere is different

We will try to stick with Kay’s notation

Probability Review Probability events: Probability that ‘event’ A

— Pr 𝐴𝐴

Probability of ‘event’ A AND B— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴,𝐵𝐵

Probability of ‘event’ A OR B— Pr 𝐴𝐴 ∪ 𝐵𝐵

Probability of an ‘event’ A, given ‘event’ B— 𝑃𝑃 𝐴𝐴|𝐵𝐵

— Pr 𝐴𝐴

Event A

Event B

<- Universe

— Pr 𝐴𝐴

Event A

Event B

<- Universe

— Pr 𝐴𝐴

Event A

Event B

<- Universe

Event A

Event B

— Pr 𝐴𝐴

Ω <- Universe

— Pr 𝐴𝐴

New UniverseΩ = B

Probability Review EXAMPLE: Probability events (fair coin flips): Probability that ‘event’ A

— Pr 𝐴𝐴

Event 𝐴𝐴 → Coin 1 is headsEvent 𝐵𝐵 → Coin 2 is heads

Probability Review EXAMPLE: Probability events (fair coin flips): Probability that ‘event’ A

— Pr 𝐴𝐴 = 1/2

Probability of ‘event’ A AND B— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴,𝐵𝐵 = 1/4

Probability of ‘event’ A OR B— Pr 𝐴𝐴 ∪ 𝐵𝐵 = 3/4

Probability of an ‘event’ A, given ‘event’ B— 𝑃𝑃 𝐴𝐴|𝐵𝐵 = 1/2

Event 𝐴𝐴 → Coin 1 is headsEvent 𝐵𝐵 → Coin 2 is heads

Probability Review Relationships between probability events: Chain Rule

— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴 Pr 𝐵𝐵|𝐴𝐴 = 𝑃𝑃 𝐵𝐵 𝑃𝑃 𝐴𝐴|𝐵𝐵

Bayes Theorem

— Pr 𝐴𝐴|𝐵𝐵 = Pr 𝐴𝐴 Pr 𝐵𝐵|𝐴𝐴Pr 𝐵𝐵

“OR” Rule— Pr 𝐴𝐴 ∪ 𝐵𝐵 = Pr 𝐴𝐴 + Pr 𝐵𝐵 − Pr 𝐴𝐴 ∩𝐵𝐵

Independence Events— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴 Pr 𝐵𝐵

Disjoint Events— Pr 𝐴𝐴 ∩ 𝐵𝐵 = 0

Bayes Theorem

New UniverseΩ = B

Weight this by probability to be in B

Bayes Theorem

New UniverseΩ = B

Derive from above

Bayes Theorem

Event A

Event B

Remove overlap

— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴 Pr 𝐵𝐵|𝐴𝐴 = Pr 𝐵𝐵 Pr 𝐴𝐴|𝐵𝐵

Bayes Theorem

Independence Events— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴 Pr 𝐵𝐵 Pr 𝐵𝐵 = Pr 𝐵𝐵|𝐴𝐴

— Pr 𝐴𝐴 ∩ 𝐵𝐵 = Pr 𝐴𝐴 Pr 𝐵𝐵|𝐴𝐴 = Pr 𝐵𝐵 Pr 𝐴𝐴|𝐵𝐵

Bayes Theorem

Event A

Event B

Probability Review Random Variables: Continuous-values random variables 𝑋𝑋

Capital-case (𝑋𝑋) means random (this is the notation we will use)

Lower-case (𝑥𝑥) means fixed value

Probability Density Functions (PDF) with parameter 𝜽𝜽 𝑝𝑝𝜃𝜃 𝑥𝑥

𝑝𝑝𝑋𝑋,𝜃𝜃 𝑥𝑥

𝑝𝑝 𝑥𝑥;𝜃𝜃

All three notations mean the same thing!

Kay’s notation

Probability Density Functions and Cumulative Density Functions

Probability Review Probability Density Functions (PDF) Definition A valid PDF is any function 𝑓𝑓 𝑥𝑥 that is both

— Non-negative p 𝑥𝑥 ≥ 0— Unit area ∫−∞

∞ 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑥𝑥 = 1

Credit: https://commons.wikimedia.org/wiki/File:Normal_Distribution_PDF.svg49

Probability Review Cumulative Density Functions (CDF) Definition A valid CDF is any function 𝐹𝐹 𝑥𝑥 that is both

— Monotonically increasing (non-deceasing )— Normalized: 𝐹𝐹 −∞ = 0, 𝐹𝐹 ∞ = 1

From a PDF as

— 𝑃𝑃 𝑥𝑥;𝜃𝜃 = Pr 𝑋𝑋 ≤ 𝑥𝑥 = ∫−∞𝑥𝑥 𝑝𝑝 𝜏𝜏;𝜃𝜃 𝑑𝑑𝜏𝜏

Figure

Credit: https://en.wikipedia.org/wiki/Normal_distribution#/media/File:Normal_Distribution_CDF.svg 50

Probability Review Gaussian (or Normal) Random Variable 𝑋𝑋:𝑁𝑁 𝜇𝜇,𝜎𝜎2

PDF is also known as the “bell curve”

mean variance

𝑝𝑝 𝑥𝑥; 𝜇𝜇,𝜎𝜎 =1

2 𝜋𝜋𝜎𝜎2exp −

𝑥𝑥 − 𝜇𝜇 2

2𝜎𝜎2

Credit:https://commons.wikimedia.org/wiki/File:Normal_Distrib

ution_PDF.svg

Probability Review Cumulative Distribution Function (CDF):

𝑃𝑃 𝑥𝑥;𝜃𝜃 = Pr 𝑋𝑋 ≤ 𝑥𝑥 = �−∞

𝑥𝑥

𝑝𝑝 𝜏𝜏;𝜃𝜃 𝑑𝑑𝜏𝜏

FigureFigure Figure

Credit: https://en.wikipedia.org/wiki/Normal_distribution#/media/File:Normal_Distribution_CDF.svg

Probability Review Example: Uniform Random Variable 𝑋𝑋: uniform 𝑎𝑎,𝑏𝑏

𝑝𝑝 𝑥𝑥;𝑎𝑎,𝑏𝑏 = �1

𝑏𝑏−𝑎𝑎for 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏

0 for otherwise Used for many applications

Credit: http://www.epixanalytics.com/modelassist/CrystalBall/Model_Assist.htm#Distributions/Continuous_distributions/Uniform.htm

Probability Review Example: Beta Random Variable 𝑋𝑋: beta(𝛼𝛼,𝛽𝛽)

𝑝𝑝 𝑥𝑥;𝛼𝛼,𝛽𝛽 = 1𝐵𝐵 𝛼𝛼,𝛽𝛽

𝑥𝑥𝛼𝛼−1 1− 𝑥𝑥 𝛽𝛽−1

Used in control systems, population genetics, Bayesian inference

Credit: https://en.wikipedia.org/wiki/Beta_distribution#/media/File:Beta_distribution_pdf.svg

Beta function

Probability Review Example: Chi-squared random variable

𝑋𝑋: 𝒳𝒳N2

𝑝𝑝 𝑥𝑥;𝑁𝑁 =𝑥𝑥𝑁𝑁2−1𝑒𝑒−

𝑥𝑥2

2𝑘𝑘2Γ 𝑁𝑁

for 𝑥𝑥 > 0

0 for otherwise

Used in detection theory

Credit: https://en.wikipedia.org/wiki/Chi-squared_distribution#/media/File:Chi-square_pdf.svg

Probability Review Transformation of Random Variables (Common Transformations)

Let 𝑥𝑥 and 𝑦𝑦 be independent such that 𝑥𝑥~𝒩𝒩 0,1 and 𝑦𝑦~𝒩𝒩 0,1

𝑥𝑥 + 𝑦𝑦~ ?

𝑥𝑥 2 ~ ?

𝑛𝑛 𝑛𝑛−1𝑛𝑛

𝑥𝑥+𝑦𝑦𝑥𝑥2+𝑦𝑦2

𝑥𝑥 2

𝑦𝑦 2 ~?

𝑥𝑥𝑦𝑦

Probability Review Transformation of Random Variables (Common Transformations)

Let 𝑥𝑥 and 𝑦𝑦 be independent such that 𝑥𝑥~𝒩𝒩 0,1 and 𝑦𝑦~𝒩𝒩 0,1

𝑥𝑥 + 𝑦𝑦~𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑎𝑎𝑛𝑛

𝑥𝑥 2 ~ 𝑐𝑐𝑐𝑐𝑐 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑛𝑛𝑠𝑠𝑑𝑑

𝑛𝑛 𝑛𝑛−1𝑛𝑛

𝑥𝑥+𝑦𝑦𝑥𝑥2+𝑦𝑦2

~ 𝑠𝑠𝑠𝑠𝑠𝑠𝑑𝑑𝑠𝑠𝑛𝑛𝑠𝑠′𝑠𝑠 − 𝑇𝑇

𝑥𝑥 2

𝑦𝑦 2 ~𝐹𝐹

𝑥𝑥𝑦𝑦

~𝐶𝐶𝑎𝑎𝑠𝑠𝑐𝑐𝑐𝑦𝑦

Expectations and Moments

Probability Review Random Variables 𝑋𝑋 <- denoted by capital letter usually

Two types: discrete, continuous

Defined by a probability distribution function (PDF) and cumulative distribution function (CDF)

For discrete-valued random variables, the PDF is replaced by a probability mass function (PMF)

Probability Review Expectation

𝐸𝐸 𝑋𝑋 = �−∞

𝑥𝑥 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑥𝑥

Expectation of a function

𝐸𝐸 𝑔𝑔 𝑋𝑋 = �−∞

𝑔𝑔 𝑥𝑥 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑥𝑥

Expectation with an unknown parameter

𝐸𝐸 𝑋𝑋;𝜃𝜃 = �−∞

𝑔𝑔 𝑥𝑥 𝑝𝑝 𝑥𝑥;𝜃𝜃 𝑑𝑑𝑥𝑥

Useful!! Can be easy to find the expectation of a function without finding find the PDF

Probability Review Moments

𝑋𝑋: 𝑁𝑁 𝑛𝑛,𝜎𝜎2𝑌𝑌 = 𝑔𝑔 𝑋𝑋 = 𝑋𝑋 −𝑛𝑛 2

— Mean: E 𝑋𝑋 = ∫−∞∞ 𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑥𝑥

— 2nd Moment: E 𝑋𝑋2 = ∫−∞∞ 𝑥𝑥2 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑥𝑥

— Variance: E 𝑋𝑋 −𝑛𝑛 2 = ∫−∞∞ (𝑥𝑥 −𝑛𝑛)2 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑥𝑥

Note: In general, the PDF of 𝑋𝑋2, 𝑋𝑋 − 𝐸𝐸 𝑋𝑋 2, … do not have the same PDF as 𝑋𝑋

Probability Review Expectation Examples Let

— 𝑋𝑋: 𝑁𝑁 𝑛𝑛,𝜎𝜎2

— 𝑌𝑌 = 𝑔𝑔 𝑋𝑋 = 𝑋𝑋 −𝑛𝑛 2

Compute

— E 𝑋𝑋;𝑛𝑛,𝜎𝜎2

— E 𝑔𝑔 𝑋𝑋 ;𝑛𝑛,𝜎𝜎2

— E 𝑋𝑋 + 10;𝑛𝑛,𝜎𝜎2

— E 𝑔𝑔 2𝑋𝑋 ;𝑛𝑛,𝜎𝜎2

— 𝑋𝑋: 𝑁𝑁 𝑛𝑛,𝜎𝜎2

— 𝑌𝑌 = 𝑔𝑔 𝑋𝑋 = 𝑋𝑋 −𝑛𝑛 2

Compute

— E 𝑋𝑋;𝑛𝑛,𝜎𝜎2 = 𝑛𝑛

— E 𝑔𝑔 𝑋𝑋 ;𝑛𝑛,𝜎𝜎2 = 𝜎𝜎2

— E 𝑋𝑋 + 10;𝑛𝑛,𝜎𝜎2 = E 𝑋𝑋;𝑛𝑛,𝜎𝜎2 + 10 = 𝑛𝑛+ 10

— E 𝑔𝑔 2𝑋𝑋 ;𝑛𝑛,𝜎𝜎2 = 4𝜎𝜎2

— 𝑋𝑋: 𝑠𝑠𝑥𝑥𝑝𝑝𝑛𝑛𝑛𝑛𝑠𝑠𝑛𝑛𝑠𝑠𝑐𝑐𝑎𝑎𝑛𝑛 𝜆𝜆

— 𝑝𝑝 𝑥𝑥;𝜆𝜆 = �𝜆𝜆𝑠𝑠−𝜆𝜆𝑥𝑥 for 𝑥𝑥 ≥ 00 for 𝑥𝑥 < 0

Compute

— E 𝑋𝑋;𝜆𝜆

Compute

— E 𝑋𝑋;𝜆𝜆 = ∫0∞𝑥𝑥 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑥𝑥 = ∫0

∞𝑥𝑥𝜆𝜆𝑠𝑠−𝜆𝜆𝑥𝑥 𝑑𝑑𝑥𝑥

E 𝑋𝑋;𝜆𝜆 = �−𝑠𝑠−𝜆𝜆𝑥𝑥 𝜆𝜆𝑥𝑥 + 1

𝜆𝜆0

=−𝑠𝑠−𝜆𝜆∞ −𝜆𝜆∞+ 1

𝜆𝜆+𝑠𝑠−0𝑥𝑥 1

𝜆𝜆

E 𝑋𝑋;𝜆𝜆 =1𝜆𝜆

L’Hospital’s Rule = 0

Compute

— E 𝑋𝑋2;𝜆𝜆

Compute

— E 𝑋𝑋2;𝜆𝜆 = ∫0∞𝑥𝑥2 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑥𝑥 = ∫0

∞𝑥𝑥2𝜆𝜆𝑠𝑠−𝜆𝜆𝑥𝑥 𝑑𝑑𝑥𝑥

E 𝑋𝑋2;𝜆𝜆 = �−𝑠𝑠−𝜆𝜆𝑥𝑥 𝜆𝜆2𝑥𝑥2 + 2𝜆𝜆𝑥𝑥 + 2

𝜆𝜆20

=−𝑠𝑠−𝜆𝜆∞ −𝜆𝜆2∞2 + 2𝜆𝜆𝑥𝑥 + 2

𝜆𝜆2 +𝑠𝑠−0𝑥𝑥 1𝜆𝜆2

E 𝑋𝑋2;𝜆𝜆 =2𝜆𝜆2

Two Random Variables (and their relationships)

Several Random Variables Joint PDFs of Random Variables 𝑝𝑝 𝑥𝑥,𝑦𝑦 Joint PDF of 2 random variables

Vector form

Define 𝑾𝑾 = 𝑋𝑋𝑌𝑌 , 𝒘𝒘 =

𝑥𝑥𝑦𝑦

𝑝𝑝 𝒘𝒘 = 𝑝𝑝 𝑥𝑥,𝑦𝑦 Joint PDF of 2 random variables (short version)

Credit: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#/media/File:MultivariateNormal.png

Several Random Variables Chain Rule for 2 Random Variables

𝑝𝑝 𝑥𝑥,𝑦𝑦 = 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑦𝑦|𝑥𝑥 = 𝑝𝑝 𝑦𝑦 𝑝𝑝 𝑥𝑥|𝑦𝑦

If Random Variables are independent

𝑝𝑝 𝑥𝑥,𝑦𝑦 = 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑦𝑦

Several Random Variables Moments of two random variables 𝑋𝑋,𝑌𝑌

— Mean: E 𝑋𝑋 , E 𝑌𝑌— 2nd Moment: E 𝑋𝑋2 ,𝐸𝐸 𝑌𝑌2

— Variance: E 𝑋𝑋 −𝑛𝑛𝑥𝑥2 , E 𝑋𝑋 −𝑛𝑛𝑦𝑦

— Cross-Correlation: E 𝑋𝑋𝑌𝑌— Cross-variance: E 𝑋𝑋 −𝑛𝑛𝑥𝑥 𝑌𝑌 −𝑛𝑛𝑦𝑦 = E 𝑋𝑋𝑌𝑌 − E 𝑋𝑋 E 𝑌𝑌

If 𝑋𝑋,𝑌𝑌 are uncorrelated Cross-Correlation: 𝐸𝐸 𝑋𝑋𝑌𝑌 = 𝐸𝐸 𝑋𝑋 𝐸𝐸 𝑌𝑌

Co-Variance: 𝐸𝐸 𝑋𝑋 − 𝐸𝐸 𝑋𝑋 𝑌𝑌 −𝐸𝐸 𝑌𝑌 = 0

Variance sum: var 𝑋𝑋 + 𝑌𝑌 = var 𝑋𝑋 + var 𝑌𝑌

Important Note: — Independent implies uncorrelated, but Correlated does not imply independent

Several Random Variables Example: Two Normal Distributions Consider the two random variables

𝑋𝑋~𝒩𝒩 0,1 , 𝑌𝑌~𝒩𝒩 𝑥𝑥, 1

Compute the joint PDF 𝑝𝑝 𝑥𝑥,𝑦𝑦

𝑝𝑝 𝑥𝑥,𝑦𝑦 = 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑦𝑦|𝑥𝑥 Chain Rule

𝑝𝑝 𝑥𝑥 = 12𝜋𝜋(1)2

exp − 𝑥𝑥 2

𝑝𝑝 𝑦𝑦|𝑥𝑥 = 12𝜋𝜋(1)2

exp − 𝑦𝑦−𝑥𝑥 2

𝑝𝑝 𝑥𝑥,𝑦𝑦 = 12𝜋𝜋(1)2

exp − 𝑥𝑥2

2(1)21

2𝜋𝜋(1)2exp − 𝑦𝑦−𝑥𝑥 2

𝑝𝑝 𝑥𝑥,𝑦𝑦 = 12𝜋𝜋

exp −𝑥𝑥2+ 𝑦𝑦−𝑥𝑥 2

2𝜋𝜋exp −2𝑥𝑥2+𝑦𝑦2−2𝑦𝑦𝑥𝑥

𝑝𝑝 𝑥𝑥,𝑦𝑦 = 12𝜋𝜋

exp −12

𝑥𝑥𝑦𝑦

𝑇𝑇 2 −1−1 1

𝑥𝑥𝑦𝑦 → 𝑋𝑋

𝑌𝑌 ~𝒩𝒩 00 , 2 −1

−1 1−1

= 1 11 2

𝑋𝑋𝑌𝑌 ~𝒩𝒩 0

0 , 1 11 2

Question: How do we interpret this distribution?

ece 6540, lecture 01

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