Exam Mistakes Flashcards
Define Nash equilibrium?
Neither player can do better by changing strategy given the other player adheres to their own strategy
Define sample space?
A list showing all possible simple outcomes that are mutually exclusive and exhaustive
Define event?
Some subset of simple outcomes from the same space
Define valid argument?
Compound proposition that come out true regardless of the truth values assigned to their single components
Explain what the εQN means mathematically?
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
When finding the critical points of a function f(x,y) how is it done?
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
What is the objective function?
The function to be maximised (ie. f(x))
What is the critical point?
The critical point of f(x) is any point x* that satisfies the equation f’(x*)=0
What is the choice variable?
The variable that the function depends on (ie. f(x) the choice variable is x)
Convex vs concave?
Convex = upward bending
ConCAVE = downward bending
How to tell if A and B are independent?
If p(A|B)=p(A)
Or
p(AB) = p(A) x p(B)
2 requirements for the total probability theorem?
If events are mutually exclusive and exhaustive
Explain the frequentists view in the Bayes debate?
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)
What do bayesians believe?
Believe there is a prior probability for any event tf use probability for a lot more events than frequentists
How to tell if two variables are IRVs?
If p(X=r,Y=s) = p(X=r) x p(Y=s)
Calculating variance of Bernoulli distribution?
Bernoulli - when 2 probabilities (Eg. assigned to 1 and 0)
Var(X)=p(1-p)
What are associative operations?
When you can put in brackets and it doesn’t change the results
What are the necessary and sufficient conditions for p->q?
p is a sufficient condition for q
q is a necessary condition for p
What is the difference between an intensive and extensive definition?
Intensive: written in notation
Extensive: fully written out
See Cartesian sets t2u2
Now
What are orthogonal vectors?
Vectors that make a right angle at the origin
2 rules of transposes?
1) (AB)^T = B^TA^T
2) (A^T)^T = A
What makes a matrix symmetric?
If it is its own transpose
Rule for a square matrix being symmetric and a normal matrix being symmetric?
For any square matrix A:
A+A^T is symmetric
For any matrix A:
A^TA and AA^T are symmetric
What makes A antisymmetric?
If A=-A^T
What is a trace?
Sum of main diagonal elements in a square matrix
Det(A) wrt transposes?
Det(A) = Det(A^T)
How to calculate det(B) when det(B) is equal to any row/column of det(A) multiplied by a scalar x?
Det(B) = xdet(A)
How to calculate det(B) when det(B) is equal to the entirety of det(A) multiplied by a scalar x?
det(B) = (x^n)det(A)
Where n is the size of the matrix A (nxn matrix)
See property 5 of determinants t2u5?
Now
What is the expansion of alien cofactors?
Expanding along a row(/column) of a matrix, choosing some other row(/column) of the matrix of cofactors, result will =0
What is singular matrix?
Where detA=0
What is an invertible function?
A function with an inverse
What does it mean if A is an invertible matrix?
The range and codomain are equal
Why do singular matrices have a zero determinant?
They lead to a loss of dimension (eg. An area mapped onto a line)
What are ill conditioned and robust matrices?
Ill conditioned: highly sensitive matrices
Robust: unsensitive matrices
What is the determinant of a diagonal matrix?
The product of its diagonal entries
A diagonal matrix has values down the main diagonal and 0s everywhere else
What makes a matrix diagonizable?
Expressing a matrix in the form:
A=XΛX^-1
And Λ is tf diagonal matrix
How to raise a diagonal matrix to any power?
Simply take the power to all of its diagonal entries
See u7t2 how to find X and Λ using eigenvectors and eigenvalues?
Now
What is a linear span?
A linear span of any set of vectors is the set of all possible linear combinations of those vectors
What is meant by ambiguity of coordinates?
See u7t2 for answer (linear independence)
What do n and k stand for in least squares regression using matrices and vectors?
n = number of observations
k = number of β
Why does taking logs of a data like population growth make sense?
Logs find proportional change tf an increase at 1% a year for example will lead to a straight line log graph
Observed series = ?
Trend + seasonal + irregular
Define range?
The image under f of the domain X (also a subset of the codomain)
How to draw a transition matrix?
FROM on the top
TO at the side
Adding DOWN the rows adds to 1.00
Define eigenvector and eigenvalue?
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
What is an intensive and extensive definition?
Intensive rights it all in notation form
Extensive will write out every possible solution
What is a Cartesian product?
(Set product)
A X B - set of ordered pairs whose first element belongs to A and second to B
What is a prisoner’s dilemma?
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
