## Description

## Question 1. (Calculus Review)

(a) Consider the function

f(x, y) = a1x

2

y

2 + 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) = a1x

2

y

2 + a2x

2

y + 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:

σ(x) = 1

1 + e−x

show that σ

0

(x) = ∂σ

∂x = σ(x)(1 − σ(x))

1

(d) Consider the following functions:

• y1 = 4x

2 − 3x + 3

• y2 = 3x

4 − 2x

3

• y3 = 4x +

√

1 − x

• y4 = x + x

−1

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:

A =

1 3 1 0 2

1 1 4 1 2

1 1 1 5 2

, b =

1

1

1

3

3

2

, AT

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(b) Consider the following objects:

A =

1 3 4

2 2 1

6 4 3

, B =

2 4

1 1

, C =

7 3 3

2 1 1

2 2 2

, D =

4 2

4 6

1 3

, u =

1

3

, v =

2

4

1

### Compute the following:

(i) AB and BA

(ii) AC and CA

(iii) AD and DA

(iv) DC and CD and DT C

(v) Bu and uB

(vi) Au

(vii) Av and vA

(viii) Av + Bv

(c) Consider the following objects:

A =

1 3 4

2 2 1

6 4 3

, u =

1

3

, v =

2

4

1

, w =

1

−2

2

.

### Compute the following:

(i) kuk1, kuk2, kuk

2

2

, kuk∞

(ii) kvk1, kvk2, kvk

2

2

, kvk∞

(iii) kv + wk1, kv + wk2, kv + wk∞

(iv) kAvk2, kA(v − w)k∞

(d) Consider the following vectors in R

2

u =

1

2

, v =

1

1

, w =

−1

1/2

Compute the dot products between all pairs of vectors. Note that the dot product may be written

using the following equivalent forms:

hx, yi = x · y = x

T

y.

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:

A =

1 3

4 1

Compute the inverse of A.

(g) Consider the 2 × 2 matrix

A =

3 3

4 4

Compute its inverse A−1

.

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

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