## Description

Homework 0: Revision

What to Submit

• A single PDF file which contains solutions to each question. For each question, provide your solution in the form of text and requested plots. For any question in which you use code, provide a copy of your code at the bottom of the relevant section.

• You are free to format your work in any way you think is appropriate. This can include using LATEX, or taking pictures of handwritten work, or writing your solutions up using a tablet. Please ensure that your work is neat, and start each question on a new page.

When and Where to Submit

• Submissions must be through Moodle – email submissions will be ignored.

Question 1. (Calculus Review) (a) Consider the function

f(x,y) = a1x2y2 + a4xy + a5x + a7

compute all first and second order derivatives of f with respect to x and y.

(b) Consider the function

f(x,y) = a1x2y2 + a2x2y + a3xy2 + a4xy + a5x + a6y + a7

compute all first and second order derivatives of f with respect to x and y.

(c) Consider the logistic sigmoid:

show that σ0(x) = ∂σ∂x = σ(x)(1 − σ(x))

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(d) Consider the following functions:

Using the second derivative test, find all local maximum and minimum points.

Question 2. (Probability Review)

(a) A manufacturing company has two retail outlets. It is known that 20% of potential customers buy products from Outlet I alone, 10% buy from both I and II, and 40% buy from neither. Let A denote the event that a potential customer, randomly chosen, buys from outel I, and B the event that the customer buys from outlet II. Compute the following probabilities:

P(A), P(B), P(A ∪ B), P(A¯B¯)

(b) Let X,Y be two discrete random variables, with joint probability mass function P(X = x,Y = y) displayed in the table below:

y

1 2 3

1 1/6 1/12 1/12

x 2 1/6 0 1/6

3 0 r 0

Compute the following quantities:

(i) r

(ii) P(X = 2,Y = 3)

(iii) P(X = 3) and P(X = 3|Y = 2)

(iv) E[X], E[Y ] and E[XY ]

(v) E[X2], E[Y 2]

(vi) Cov(X,Y )

(vii) Var(X) and Var(Y )

(viii) Corr(X,Y )

(ix) E[X + Y ], E[X + Y 2], Var(X + Y ) and Var(X + Y 2).

Question 3. (Linear Algebra Review)

(a) Write down the dimensions of the following objects:

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(b) Consider the following objects:

Compute the following:

(i) AB and BA

(ii) AC and CA

(iii) AD and DA

(iv) DC and CD and DTC

(v) Bu and uB

(vi) Au

(vii) Av and vA

(viii) Av + Bv

(c) Consider the following objects:

.

Compute the following:

(i) kuk1,kuk2,kuk22,kuk∞

(ii) kvk1,kvk2,kvk22,kvk∞

(iii) kv + wk1,kv + wk2,kv + wk∞

(iv) kAvk2,kA(v − w)k∞

(d) Consider the following vectors in R2

hx,yi = x · y = xTy.

Then compute the angle between the vectors and plot.

(e) Dot products are extremely important in machine learning, explain what it means for a dot product to be zero, positive or negative. (f) Consider the 2 × 2 matrix:

Compute the inverse of A.

(g) Consider the 2 × 2 matrix

Compute its inverse A−1.

(h) Let X be a matrix (of any dimension), show that XTX is always symmetric.

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