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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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