EMTH210 final

Two conditions for a PDF

nonnegative everywhere and total area under the curve = 1

shifting has no impact on….?

variance

bigger sample =

smaller variance

When two random variables are independent, the joint PDF factors into ….?

the product of its marginal PDFs (lowercase f)

When X,Y are independent, the Joint Cumulative Distribution Function (JCDF) factors into…?

the product of their individual marginal Cumulative Distribution Functions (CDFs) (capital F)

for eigenvectors to be orthogonal (perpendicular) to one another:

their dot product must equal 0.

what is the characteristic equation

det(A-lambdaI) = 0

when asked to find the general solution it means…

find x (or x and y)

How to find the eigenvalues of a diagonal matrix

the values along the diagonal are the eigenvalues

for a symmetric matrix, eigenvectors are….?

orthogonal

If a matrix is orthogonal and mulitplied together by its transpose what is the product?

the identity matrix

integration by parts formula

integral of u dv = uv - integral of v du

requirements for multivariate iterated integrals (in limits and variables)

the outermost limits must be constants, inner limits can only depend on outer variables, and you cannot have the same differential variable repeated.

How do you mathematically prove whether two continuous random variables X and Y are independent using their probability density functions?

They are independent if and only if the product of their marginal PDFs equals their joint PDF for every x and y

For n independent random variables each with mean mu and variance sigma squared, what are the expected value and variance of the sample average X(overline)

expected value = mu , variance = variance / n

In Maximum Likelihood Estimation (MLE), how is the log-likelihood function mathematically constructed from a sample of independent observations?

It is the natural logarithm of the product of the individual probability density functions, which simplifies into the sum of the log-PDFs

What is the first major calculus step used to find a candidate value for a maximum likelihood estimator theta hat

You differentiate the log-likelihood function with respect to the parameter theta, set this first derivative equal to zero, and solve for theta

Conceptually, what are you doing when you change the integration order of a double integral from dydx to dxdy

Changing the direciton of the arrows across the region

If two random variables are independent, their covariance is 0. If their covariance is 0, are they guaranteed to be independent? Explain.

Covariance only detects linear relationships. Two variables can have a covariance of 0 but remain completely dependent through a non-linear relationship (such as Y = X²)

Holding your collected sample dataset completely fixed, why does demanding a higher level of confidence (e.g., 90% to 95%) force your confidence interval to become wider? (to do with mean-target zone)

A wider interval provides a broader target zone, lowering the risk that the true mean falls outside your boundaries. Precision is sacrificed to gain certainty.

ODE: Given a system dy/dt = By with distinct eigenvalues lambda1, lambda2, lambda3 and corresponding eigenvectors v1, v2, v3:

What is the General Solution Formula for the state vector y(t)?

c1*e^(lambda1*t)*v1 + c2*e^(lambda2*t)*v2 + c3*e^(lambda3*t)*v3

matrix form of quadratic equation (ellipse)

x^TAx

density = (formula)

mass / volume

generally, how do you find the expectation of a function? for any number of variables

integration of the function times the inside of the E() brackets, with respect to however many variables present

what is the likelihood function of a function with independent variables

The product of the individual probability density functions for each observed value, treated as a function of the unknown parameter

how do you simplify the log-likelihood function? (after adding ln)

convert the multiplications / products into summations

what is x (overline)

the sum all observed values of x divided by the number of observations (samples)

what does correlation show and how is it different to covariance? (and what scale)

correlation tells you both the direction and strength while covariance only shows you direction. It scales the relationship to a clean, unit-free metric between -1 and 1.

formula for centroid of shape in 2 space

1/A double integral

method steps in defining new coordinates u and v and writing ellipse in form au²+bv²=5

write ellipse as matrix form, xAx^T. Replace A = PDP^T. Let [u v] = [x y]P. write in new form to find coefficients.

integral for coordinates of centroid (x or y)

1/A integral of (x or y)

expectation and variance of exponential random variable

1/lambda and 1/lambda²

expectation and variance of a continuous uniform random variable

(1/2)(a+b) and (1/12)(b-a)²

How do you find the variance of a linear combination of two random variables when they are independent? what is Var(aX + bY + c)

a²Var(X)+b²Var(Y)

How do you find the variance of a linear combination of two random variables when they are not independent? what is Var(aX + bY + c)

a²Var(X)+b²Var(Y)+2abCov(X,Y)

sample variance formula s²=

1/(n-1) sum up to n of (xi - sample mean)²

sample mean formula x(overline) =

1/n sum up to n of x