Calculus and complex analysis
The derivative of the function x2
What is 2x?
A m by n matrix can only be multiplied by a matrix with this many rows
The mean of the numbers 2, 4, 6, 8, 10
What is 6?
The maximum value a probability measure can be
What is 1?
A collection of things
What is a set?
The limit definition of a derivative
What is the limit as h approaches 0 of (f(x+h)-f(x))/h?
The roots of this give you the eigenvalues for a matrix
What is the characteristic polynomial?
The odds of rolling a 23 if you roll two fair 45 sided die
What is 2/45?
The most commonly used distribution and its parameters
What is the normal distribution and the mean and standard deviation?
A proof where you prove the base case first then use that to show that it is true for all natural numbers.
What is proof by induction?
The integral of sec2(x)
What is tan(x)+C
A set of vectors much be these two things to be a basis for a vector space.
What is spanning the vector space and linear independence?
The meaning of the correlation between two variables being 1
What is a linear relationship between the variables?
The formula to find P(B|A) if you know P(A),P(B),P(A|B)
What is P(B∣A)=P(A|B)*P(B)/P(A)?
7+9 in Z6 or the integers modulo 6.
What is 4?
A complex function sastifies this to have a derivative.
What are the Cauchy-Riemann equations
A geometric interpretation of eigenvalues
The eigenvalue is the scalar by which the eigenvector is stretched or compressed by when multiplied by its matrix
What is an error that occurs when you reject a null hypothesis that is actually true and an error that occurs when you fail to reject a null hypothesis that is actually false
The difference between a discrete probability distribution and a continuous one.
What is discrete distribution is countable, while the continuous one is uncountable?
The most general form of an algebraic structure with two operations, addition and multiplication.
What is a ring?
A function which is analytic on the entire complex plane
What is an entire function?
Any symmetric matrix, X, can be decomposed into PTAP where P is orthogonal, and A is diagonal where the elements are the eigenvalues of X
What is the Spectral Theorem?
A technique in statistics that allows us to estimate the sampling distribution of a statistic by repeatedly resampling, with replacement, from the observed data
What is bootstrapping?
This theorem states that for a large enough sample size of any random variable, the distribution of the sample will be approximately normal
What is the central limit theorem (CLT)?
A field which contains as an element every root of every polynomial definable over said field and an example of one such field.
What is algebraically closed, and the complex numbers?