SI&DA - Lecture 2 'Recap of Random Variables (RVs) - Part II'

Recap of Lecture 1

→ SI&DA - Lecture 1 ‘Recap of Random Variables (RVs) - Part I’

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SI&DA - Theorem 'Affine RV'

Theorem ‘Affine RV’

SIandDATheoremDemonstration

Where: → $m$ is the mean of $X$ → $P$ is the variance of $X$

NOTE: This is also true for $Y=AX+b$ even if $X$ is not Gaussian, still the affine RV $Y$ will be a Gaussian Distribution

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Dim:

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SI&DA - Definition of 'Cross-Covariance'

Definition of ‘Cross-Covariance’

SIandDADefinition

→ SI&DA - Mean & Variance

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SI&DA - Property of Multivariate Gaussian RVs

Property of Multivariate Gaussian RVs

SIandDATheorem

→ SI&DA - Multivariate Distributions → SI&DA - Gaussian Random Variables

Given $X$ and $Y$ such that:

Where: $P_x:=$ Variance of $x$ $P_y:=$ Variance of $y$ $P_{xy}:=$ Cross-Covariance of $x$ and $y$ $P_{yx}:=$ Cross-Covariance of of $y$ and $x$

→ Then:

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SI&DA - Central Limit Theorem

Central Limit Theorem

SIandDATheorem

Other Definition: DES - Central Limit Theorem

→ SI&DA - Definition of ‘Independent Random Variables’ → SI&DA - Mean & Variance → SI&DA - Gaussian Random Variables

RESULT:

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~Ex.: Homework (1)

SIandDAHomework

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SI&DA - Functions of RVs

Functions of RVs

SIandDADefinitionExample

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~Ex.:

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~Ex.:

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SI&DA - Multivariate Functions of RVs

Multivariate Functions of RVs

SIandDADefinition

→ SI&DA - Multivariate Functions of RVs (Linear Case)

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SI&DA - Multivariate Functions of RVs (Linear Case)

Multivariate Functions of RVs (Linear Case)

SIandDAExample

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~Ex.:

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~Ex.: Homework (2)

SIandDAHomework