Linear Algebra
A set of vectors that are orthogonal and each have length 1.
What is an orthonormal set of vectors?
An ODE where the independent variable does not appear explicitly.
What is the definition of an autonomous ODE?
Define infinite series.
For a real number an , an infinite series is an expression of the form a1 + a2 + a3 + … + an + …
P(X1=x1 , X2=x2 , ... , Xn=xn | θ)
What is the Likelihood Function?
The set of vectors v such that Av = 0.
What is the nullspace of a matrix A?
V = ∫ F(x) dx
What is the potential energy V(x) for any (integrable) scalar function F(x) for motion along a line?
There is no real number s such that sn -> s as n → ∞ .
What is the definition of a divergent series?
Let X be a random variable with mean µ and variance σ2 . If X̄n is the mean of a random sample of size n drawn from the distribution of X, then the distribution of the statistic (X̄n−µ)/(σ/√n) tends to the standard normal distribution as n → ∞.
What is the Central Limit Theorem?
If matrix B is obtained by adding a multiple of a row or column of matrix A to another row or column, then what is the relationship between the determinants of A and B?
det(A)=det(B)
Sometimes described as the "quantity of motion", this vector quantity is calculated as mv.
What is momentum?
The function f : [1 , ∞] -> R must be: 1. non-negative 2. decreasing 3. continuous
What are the conditions for the Integral Test?
[ (X̄1-X̄2) - zα/2√(σ12/n1 + σ22/n2) , (X̄1-X̄2) + zα/2√(σ12/n1 + σ22/n2) ]
What is a confidence interval?
How do you find the eigenvalues of a matrix?
det(A − λIn)=0
This field has strength g and is usually represented as -gj where j is a unit vector in the orthonormal basis {i, j, k}
What is the gravitational field?
If 0 ≤ an ≤ bn for all n∈N and b1 + b2 + b3 + … + bn + … converges, then a1 + a2 + a3 + … + an + … converges.
What is the Comparison Test?
Let Z1,Z2,...,Zn be i.i.d. standard normal variables. Then Y=Σi=1nZi has what distribution?
Hint: This distribution has k degrees of freedom.
What is the Χ2 distribution?
What is the operation of this matrix on a vector?
A = [ cosθ -sinθ ] [ sinθ cosθ ]
Rotation.
x(t) = x0(t) + εx1(t) + ε2x2(t) + ...
What is a perturbation expansion?
If a1 + a2 + a3 + … + an + … is a convergent series, then (an)n∈N is a null sequence.
What is the nth term test?
This is the probability that the random end-points of I(X) contain the true value of θ.
What is the coverage probability of an interval estimator for θ?