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1st Year Maths > Exam Mistakes > Flashcards

Exam Mistakes Flashcards

1
Q

Define Nash equilibrium?

A

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

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2
Q

Define sample space?

A

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

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3
Q

Define event?

A

Some subset of simple outcomes from the same space

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4
Q

Define valid argument?

A

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

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5
Q

Explain what the εQN means mathematically?

A

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

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6
Q

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

A

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

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7
Q

What is the objective function?

A

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

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8
Q

What is the critical point?

A

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

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9
Q

What is the choice variable?

A

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

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10
Q

Convex vs concave?

A

Convex = upward bending

ConCAVE = downward bending

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11
Q

How to tell if A and B are independent?

A

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

Or

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

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12
Q

2 requirements for the total probability theorem?

A

If events are mutually exclusive and exhaustive

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13
Q

Explain the frequentists view in the Bayes debate?

A

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)

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14
Q

What do bayesians believe?

A

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

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15
Q

How to tell if two variables are IRVs?

A

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

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16
Q

Calculating variance of Bernoulli distribution?

A

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

Var(X)=p(1-p)

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17
Q

What are associative operations?

A

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

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18
Q

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

A

p is a sufficient condition for q

q is a necessary condition for p

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19
Q

What is the difference between an intensive and extensive definition?

A

Intensive: written in notation

Extensive: fully written out

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20
Q

See Cartesian sets t2u2

A

Now

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21
Q

What are orthogonal vectors?

A

Vectors that make a right angle at the origin

22
Q

2 rules of transposes?

A

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

2) (A^T)^T = A

23
Q

What makes a matrix symmetric?

A

If it is its own transpose

24
Q

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

A

For any square matrix A:

A+A^T is symmetric

For any matrix A:

A^TA and AA^T are symmetric

25
Q

What makes A antisymmetric?

A

If A=-A^T

26
Q

What is a trace?

A

Sum of main diagonal elements in a square matrix

27
Q

Det(A) wrt transposes?

A

Det(A) = Det(A^T)

28
Q

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

A

Det(B) = xdet(A)

29
Q

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

A

det(B) = (x^n)det(A)

Where n is the size of the matrix A (nxn matrix)

30
Q

See property 5 of determinants t2u5?

A

Now

31
Q

What is the expansion of alien cofactors?

A

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

32
Q

What is singular matrix?

A

Where detA=0

33
Q

What is an invertible function?

A

A function with an inverse

34
Q

What does it mean if A is an invertible matrix?

A

The range and codomain are equal

35
Q

Why do singular matrices have a zero determinant?

A

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

36
Q

What are ill conditioned and robust matrices?

A

Ill conditioned: highly sensitive matrices

Robust: unsensitive matrices

37
Q

What is the determinant of a diagonal matrix?

A

The product of its diagonal entries

A diagonal matrix has values down the main diagonal and 0s everywhere else

38
Q

What makes a matrix diagonizable?

A

Expressing a matrix in the form:

A=XΛX^-1

And Λ is tf diagonal matrix

39
Q

How to raise a diagonal matrix to any power?

A

Simply take the power to all of its diagonal entries

40
Q

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

A

Now

41
Q

What is a linear span?

A

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

42
Q

What is meant by ambiguity of coordinates?

A

See u7t2 for answer (linear independence)

43
Q

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

A

n = number of observations

k = number of β

44
Q

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

A

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

45
Q

Observed series = ?

A

Trend + seasonal + irregular

46
Q

Define range?

A

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

47
Q

How to draw a transition matrix?

A

FROM on the top
TO at the side

Adding DOWN the rows adds to 1.00

48
Q

Define eigenvector and eigenvalue?

A

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

49
Q

What is an intensive and extensive definition?

A

Intensive rights it all in notation form

Extensive will write out every possible solution

50
Q

What is a Cartesian product?

A

(Set product)

A X B - set of ordered pairs whose first element belongs to A and second to B

51
Q

What is a prisoner’s dilemma?

A

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

1st Year Maths (1 decks)

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