Exam Mistakes

Question 1

Define Nash equilibrium?

Answer 1

Neither player can do better by changing strategy given the other player adheres to their own strategy

Question 2

Define sample space?

Answer 2

A list showing all possible simple outcomes that are mutually exclusive and exhaustive

Question 3

Define event?

Answer 3

Some subset of simple outcomes from the same space

Question 4

Define valid argument?

Answer 4

Compound proposition that come out true regardless of the truth values assigned to their single components

Question 5

Explain what the εQN means mathematically?

Answer 5

Suppose that N changes by a small proportion h, and that as a result changes the result Q by a proportion k. Then εQN is approximately k/h. The smaller h (and consequently k) gets, the closer k/h gets to εQN

Question 6

When finding the critical points of a function f(x,y) how is it done?

Answer 6

Differentiate wrt to x and then to y

Make both equations equal zero

Find PAIRS of values for x and y that make both equations equal zero

Question 7

What is the objective function?

Answer 7

The function to be maximised (ie. f(x))

Question 8

What is the critical point?

Answer 8

The critical point of f(x) is any point x* that satisfies the equation f’(x*)=0

Question 9

What is the choice variable?

Answer 9

The variable that the function depends on (ie. f(x) the choice variable is x)

Question 10

Convex vs concave?

Answer 10

Convex = upward bending

ConCAVE = downward bending

Question 11

How to tell if A and B are independent?

Answer 11

If p(A|B)=p(A)

Or

p(AB) = p(A) x p(B)

Question 12

2 requirements for the total probability theorem?

Answer 12

If events are mutually exclusive and exhaustive

Question 13

Explain the frequentists view in the Bayes debate?

Answer 13

When using Bayes theorem to calculate p(C|F), frequentists say it only makes sense to assign a value to p(C) if it is a proportion

Why?
They believe the concept of probability only applies if there is numerical evidence available (ie. Doesn’t occur for ‘one off’ events)

Question 14

What do bayesians believe?

Answer 14

Believe there is a prior probability for any event tf use probability for a lot more events than frequentists

Question 15

How to tell if two variables are IRVs?

Answer 15

If p(X=r,Y=s) = p(X=r) x p(Y=s)

Question 16

Calculating variance of Bernoulli distribution?

Answer 16

Bernoulli - when 2 probabilities (Eg. assigned to 1 and 0)

Var(X)=p(1-p)

Question 17

What are associative operations?

Answer 17

When you can put in brackets and it doesn’t change the results

Question 18

What are the necessary and sufficient conditions for p->q?

Answer 18

p is a sufficient condition for q

q is a necessary condition for p

Question 19

What is the difference between an intensive and extensive definition?

Answer 19

Intensive: written in notation

Extensive: fully written out

Question 20

See Cartesian sets t2u2

Answer 20

Now

Question 21

What are orthogonal vectors?

Answer 21

Vectors that make a right angle at the origin

Question 22

2 rules of transposes?

Answer 22

1) (AB)^T = B^TA^T

2) (A^T)^T = A

Question 23

What makes a matrix symmetric?

Answer 23

If it is its own transpose

Question 24

Rule for a square matrix being symmetric and a normal matrix being symmetric?

Answer 24

For any square matrix A:

A+A^T is symmetric

For any matrix A:

A^TA and AA^T are symmetric

Question 25

What makes A antisymmetric?

Answer 25

If A=-A^T

Question 26

What is a trace?

Answer 26

Sum of main diagonal elements in a square matrix

Question 27

Det(A) wrt transposes?

Answer 27

Det(A) = Det(A^T)

Question 28

How to calculate det(B) when det(B) is equal to any row/column of det(A) multiplied by a scalar x?

Answer 28

Det(B) = xdet(A)

Question 29

How to calculate det(B) when det(B) is equal to the entirety of det(A) multiplied by a scalar x?

Answer 29

det(B) = (x^n)det(A) Where n is the size of the matrix A (nxn matrix)

Question 30

See property 5 of determinants t2u5?

Answer 30

Now

Question 31

What is the expansion of alien cofactors?

Answer 31

Expanding along a row(/column) of a matrix, choosing some other row(/column) of the matrix of cofactors, result will =0

Question 32

What is singular matrix?

Answer 32

Where detA=0

Question 33

What is an invertible function?

Answer 33

A function with an inverse

Question 34

What does it mean if A is an invertible matrix?

Answer 34

The range and codomain are equal

Question 35

Why do singular matrices have a zero determinant?

Answer 35

They lead to a loss of dimension (eg. An area mapped onto a line)

Question 36

What are ill conditioned and robust matrices?

Answer 36

Ill conditioned: highly sensitive matrices Robust: unsensitive matrices

Question 37

What is the determinant of a diagonal matrix?

Answer 37

The product of its diagonal entries A diagonal matrix has values down the main diagonal and 0s everywhere else

Question 38

What makes a matrix diagonizable?

Answer 38

Expressing a matrix in the form: A=XΛX^-1 And Λ is tf diagonal matrix

Question 39

How to raise a diagonal matrix to any power?

Answer 39

Simply take the power to all of its diagonal entries

Question 40

See u7t2 how to find X and Λ using eigenvectors and eigenvalues?

Answer 40

Now

Question 41

What is a linear span?

Answer 41

A linear span of any set of vectors is the set of all possible linear combinations of those vectors

Question 42

What is meant by ambiguity of coordinates?

Answer 42

See u7t2 for answer (linear independence)

Question 43

What do n and k stand for in least squares regression using matrices and vectors?

Answer 43

n = number of observations k = number of β

Question 44

Why does taking logs of a data like population growth make sense?

Answer 44

Logs find proportional change tf an increase at 1% a year for example will lead to a straight line log graph

Question 45

Observed series = ?

Answer 45

Trend + seasonal + irregular

Question 46

Define range?

Answer 46

The image under f of the domain X (also a subset of the codomain)

Question 47

How to draw a transition matrix?

Answer 47

FROM on the top TO at the side Adding DOWN the rows adds to 1.00

Question 48

Define eigenvector and eigenvalue?

Answer 48

Suppose there is a scalar value λ(set of real numbers) and a NONZERO nx1 vector x sic that Ax=xλ, then x is said to be an eigenvector of A, and λ is the corresponding eigenvalue

Question 49

What is an intensive and extensive definition?

Answer 49

Intensive rights it all in notation form Extensive will write out every possible solution

Question 50

What is a Cartesian product?

Answer 50

(Set product) A X B - set of ordered pairs whose first element belongs to A and second to B

Question 51

What is a prisoner's dilemma?

Answer 51

A paradox in decision analysis in which two individuals acting in their own self interest pursue a course of action that does not result the ideal outcome