2) Background Flashcards by Caleb Pleavin

Q

How is the inner (dot) product and the Euclidean norm of vectors in Rn defined

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Q

What is the sign function

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Q

What is the indicator function

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Q

How do we define a symmetric, positive semi-definite, and positive definite matrix

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Positive semi-definite implies all eigenvalues are ≥ 0

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Q

How do we define probability mass functions and probability density functions

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Q

What is the normal distribution

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Q

What is the Gaussian distribution

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Q

What is the Binomial Distribution

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Q

What is the Uniform Distribution

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Q

How is the expectation of a function defined

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Q

How do we define joint and marginal distributions for random variables

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Q

What are independent random variables

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Q

What is conditional probability

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Q

How do we define joint densities, marginal densities, and conditional densities for random variables X and Y

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Q

What is covariance

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Q

What is a covariance matrix

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Q

What is the Strong Law of Large Numbers

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Q

What is the Weak Law of Large Numbers

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Q

What are Lagrange polynomials

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Q

What is the error formula for interpolation using Lagrange polynomials

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Q

What is Big-O notation

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Q

What is convex set

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Q

What are examples of convex sets

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Q

What is the convex hull of a set

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A

Convex Hull - the intersection of all
convex sets containing B as a subset

What can be said about the convex hull of a finite set B

If B ⊆Rn and is a finite set, then conv(B) is a polyhedron

How is a convex function defined on a convex set A

How can convexity of a function f:A→R be characterised using derivatives

What are some examples of convex functions

What is the space of continuous functions C(X), and how is convergence defined in this space

What does it mean for a set C^ to be dense in C(X)

A set C^ is dense in C(X) if for every f∈C(X) and every ε>0, there exists g∈ C^ such that ∥f−g∥∞ <ε; meaning, functions in C^ can approximate any continuous function in C(X) to any desired accuracy

What is Weierstrass Theorem

Why does the space of polynomials PK of fixed degree not give density in C([0,1]), and what is the significance of the Weierstrass theorem