Exam Prep

Question 1

Pairwise Independent

Answer 1

All pairs in a set are equal in probability.
P(A1)P(A2) = P(A1 n A2).

Question 2

Mutual Independence

Answer 2

Mutual independence is everything is independent together plus the pairwise independence.

Question 3

Normal Distribution: increase μ in pdf

Answer 3

Shifts normal distribution by distance μ in the positive direction.

Question 4

Normal Distribution: decrease μ in pdf

Answer 4

Shifts normal distribution by distance μ in the negative direction.

Question 5

Normal Distribution: increase σ in pdf

Answer 5

Stretches the distribution and
lowers the maximum of the function.

Question 6

Normal Distribution: decrease σ in pdf

Answer 6

Narrows the distribution and
heightens the maximum of the function.

Question 7

Reflexive

Answer 7

a = a.

Question 8

Symmetric

Answer 8

a, b = b, a.

Question 9

Transitive

Answer 9

a, b and b, c implies a, c.

Question 10

Sample Standard Deviation

Answer 10

sqrt(1/n-1 x (sum(x - meanx)^2))

Question 11

Sample Pearson Correlation

Answer 11

1/n-1 x ((sum(x-meanx)(y-meany)) / sx x sy)
where
sx = sample standard deviation of x
sy = sample standard deviation of y

Question 12

Isometry Matrix

Answer 12

An isometry is any map f : R^m -> R^m that is defined on the whole space R^m and preserves the Euclidean L_2-metric.

Question 13

Isometric Transformations

Answer 13

Reflection
Rotation
Translation
Orthogonal transformation

Question 14

Orthogonal Transformation

Answer 14

A transformation with preserves a symmetric inner product.
Aka, it preserves lengths of vectors and angles between vectors.

Question 15

Reduce dimensionality

Answer 15

• SVD
• PCA
• (if two-dimensional) linear regression methods

Question 16

Standarised z-score

Answer 16

X - μ / σ
X = random variable
σ = standard deviation
μ = mean

Question 17

Significance Level

Answer 17

x - μ / σ - sqrt(n)
x = given number (like an estimated mean)
σ = standard deviation
μ = mean
m = amount of data given (like amount of employees etc)

Question 18

Variance

Answer 18

σ^2
E(X^2) - (E(X))^2

Question 19

Standard Deviation

Answer 19

sqrt((sum(x - meanx)^2)/N)

Question 20

Sample Variance

Answer 20

(sum(x - meanx)^2)/N-1

Question 21

Covariance

Answer 21

E(XY) - (EX)(EY)
1/N x sum(x-meanx)(y-meany)

Question 22

Sample covariance

Answer 22

1/N-1 x sum(x-meanx)(y-meany)

Question 23

Pearson product-moment correlation

Answer 23

cov(X,Y) / (σ_X)(σ_Y)

Question 24

Size of a vector

Answer 24

|u|
sqrt(all values ^2 and then added together).

Question 25

Angle between vectors

Answer 25

angle = arccos(dot(A,B) / (|A|* |B|))

Question 26

Scalar Product

Answer 26

Let vector a = {1,2,3}
and vector b = {4,5,6}
Then a . b = (1 x 4) + (2 x 5) + (3 x 6)

Question 27

Finding orthogonal vectors

Answer 27

Lets say we have u = {1,2,3}
To find an orthogonal vector v to u, it must satisfy:
a) u . v = 0, or as rewrote by b,
b) a + 2b + 3c = 0 (example specific) where v = {a,b,c}
In this instance, we can have v = {0, -3, 2} since 0 - 6 + 6 = 0.
If we then apply a, we get:
(1 x 0) + (2 x -3) + (3 x -2) = 0

Question 28

How to find if matrix is orthogonal, and therefore isometric

Answer 28

For matrix a:
a)
if A^T x A = I
or in plain text, if the transpose of the matrix multiplied by the matrix equals the identity matrix.
A way to figure this out is by checking if the columns of A are length 1.
b)
preserves scalar products.
We are just now checking if the columns are orthogonal to each other.
Therefore, we multiply the columns together, and if they all equal 0, then as long as a is also fulfilled, we have an isometry.

Question 29

Sample correlation coefficient

Answer 29

sum((x-meanx)(y-meany)) / sqrt(sum((x-meanx)^2) x (sum((y-meany)^2))

Question 30

Proving something is a metric

Answer 30

• distance is never negative, and the only way to have 0 distance is to point to itself
• distance is symmetric
• triangle inequality holds (a + b >= c)

Question 31

Single Linkage

Answer 31

Merge trees based upon minimum difference between text.

Question 32

Complete Linkage

Answer 32

Merge trees based upon maximum difference between text.

Question 33

Hamming Distance

Answer 33

Binary difference.

Question 34

Manhattan Metric

Answer 34

Denoted as L_1(p, q), and is calculated by:
sum(|p_i - q_i|)

Question 35

Euclidean Metric

Answer 35

Denoted as L_2, and is calculated by:
sqrt(sum(p_i - q_i)^2)

Question 36

Max Matric

Answer 36

Denoted as L_∞, and is calculated by:
max (|p_i - q_i|

Question 37

Is matrix a bijective?

Answer 37

If det(a) != 0, then matrix a is bijective.

Question 38

Det() for 2x2

Answer 38

ad - bc

Question 39

Det() 3x3

Answer 39

a11a22a33 + a12a23a32 + a13a21a32 - a13a22a31 - a12a21a33 - a11a21a32

Question 40

Det() signed area of a triangle

Answer 40

Coordinates of triangle - (x1, y1) (x2, y2) (x3, y3)
1/2 x det(
x2 - x1 x3 - x1
y2 - y1 y3 - y1
)

Question 41

Convex Hull

Answer 41

The shape of the smallest convex set that contains it.

Question 42

Convex Set

Answer 42

A subset that intersects every line into a single line segment.

Question 43

Invariant

Answer 43

Lets say you have apples. If we say “if it is an apple, then it is a fruit”, then we can say that fruit is an invariant. Hence, invariants are an true if statement.

Question 44

Complete invariant

Answer 44

Saying “if A is an apple, then it is produced from apple trees” is a complete invariance statement, because then we can say “if it is produced from an apple tree, then it is an apple” which is also a true statement.

Question 45

Trace of a matrix

Answer 45

All diagonal elements summed.
If we had:
1 2 3
4 5 6
7 8 9
Then the trace would be 1 + 5 + 9.

Question 46

Eigenvalue

Answer 46

If v satisfies:
Av = λv
then v is an eigenvalue.
A is the matrix, v is the eigenvector, λ is the eigenvalue.

Question 47

Det() with eigenvalues

Answer 47

det(A - λI) = 0
where A is the matrix, λ is the eigenvalue, I is the identity matrix.

Question 48

Finding eigenvectors / eigenvalues

Answer 48

From matrix a, we can
1) find eigenvalues by finding the determinant of the matrix with every diagonal element subtracted by lambda, which needs to equal 0.
2) find eigenvectors as part of a solution of (A - λI) = 0.

Question 49

Change in linear basis

Answer 49

Lets say we had a basis RB which has v1 = 3e1 + e2, v2 = -2e1 + e2.
If we wanted to translate a specific vector, lets say (2, 1), from RB to GB, we would:
1) make the equation for the vector 2v1 + v2.
2) 2v1 + v2 = (3e1 + e2) + (-2e1 + e2)
3) multiply 2v1 and 3e1, and 2v1 and e2 to give us 6e1 + 2e2.
4) multiply v2 and -2e1, and v2 and e2 giving us -2e1 + e2.
This results in (6e1 - 2e1) + (2e2 + e2) = 4e1 + 3e2

Question 50

Applying linear map with linear basis

Answer 50

1) translate RB coordinates to GB : B_rv.
2) apply the map GB : AB_rv
3) translate result back from GB to RB : (B^-1 AB)rv.

Question 51

Steps of PCA

Answer 51

1) subtract means from respective values
2) find the covariance matrix by creating a matrix S consisting of results of x being on the top or left, and results of y being on the bottom or right. We then apply the formula
SS^T / N-1.
3) find eigenvalues and eigenvectors.
4) sort the eigenvectors.
5) pick the top eigenvectors and transform the normalised dataset by multiplication

Question 52

Multiply 2x2 matrix

Answer 52

row1column1 row1column2
row2column1 row2column2

Question 53

Multiply 2x3 matrix

Answer 53

row1column1 row1column2
row2column1 row2column2

Question 54

Multiply 3x3 matrix

Answer 54

row1column1 row1column2 row1column3
row2column1 row2column2 row2column3
row3column1 row3column2 row3column3

Question 55

Inverse of 2x2

Answer 55

• swap a and d
• negative b and c
• divide every number by det(original)