Exam Flashcards
(22 cards)
What is the goal of Calculus of Variations?
To find a continuously differentiable function y(x) that minimizes or maximizes an integral of the form ∫ from x1 to x2 of F(x, y, y’) with respect to x1.
The integral is referred to as a functional.
Who originated the technique of Calculus of Variations?
Isaac Newton, James Bernoulli, John Bernoulli, Euler, and Lagrange.
Euler and Lagrange independently determined the fundamental equation.
What is the fundamental equation in Calculus of Variations?
∂F/∂y - d/dx(∂F/∂y’) = 0.
This equation is named after Euler and Lagrange.
What does the Time Value of Money concept entail?
It includes future value and present value.
Present value means reducing the value of a future sum of money to find out how much it is worth today by discounting.
What is the formula for Simple Interest?
Principal times interest rate times time.
This is a basic calculation for interest earned on a principal amount.
What is the difference between Simple Interest and Compound Interest?
Simple Interest is earned only on the principal, while Compound Interest is earned on both the principal and accumulated interest.
Compounding can occur at various intervals such as annually or continuously.
What is Net Present Value (NPV)?
The difference between the market value of a project and the cost incurred.
It is computed by finding the present value of each cash flow and summing them up, subtracting the initial investment.
What does a positive NPV indicate?
The project is expected to add value to the firm and should be accepted.
NPV accounts for the time value of money and the risk of cash flows.
What is the Internal Rate of Return (IRR)?
The discount rate that makes the NPV of a project equal to zero.
It represents the project’s expected return.
What is an Annuity?
An asset that pays a fixed sum at each equal period of time for a pre-specified and finite number of years.
Characteristics include regular, equal payments, generally at a fixed rate of interest.
What distinguishes an Ordinary Annuity from an Annuity Due?
Ordinary annuity payments are made at the end of each period, while annuity due payments are made at the beginning of each period.
This affects the present value calculations of the annuities.
What is the purpose of the Laplace transform?
To solve difference equations.
This technique transforms complex equations into a simpler form for easier analysis.
What is the Central Limit Theorem (CLT)?
It states that the variance of sample means is the population variance divided by the sample size.
This implies the standard error is σ/√n.
What does Ordinary Least Squares (OLS) aim to achieve?
To fit a line to data by minimizing the sum of squared errors between observed data points and predicted values.
It’s a fundamental technique in econometrics.
What is a Type I Error in hypothesis testing?
Rejecting the null hypothesis when it is true.
The probability of a Type I error is known as the significance level (alpha, α).
What is a confidence interval?
An upper and lower bound within which a population parameter is thought to lie, with a certain level of confidence.
For example, a 95% confidence interval means there is a 95% chance the parameter falls within that range.
What is the formula for calculating variance?
A measure of dispersion.
The standard deviation is the square root of variance and is often more useful.
What is the significance of the degrees of freedom in statistics?
For a sample variance, we use n-1 degrees of freedom.
This accounts for the fact that one sample value is determined by the others.
What is the formula for calculating depreciation?
(initial value - scrap value) / estimated lifespan.
This formula estimates the annual decline in asset value over time.
What is the difference between Unconstrained and Constrained Optimization?
Unconstrained optimization has no restrictions on variables, while constrained optimization involves optimizing an objective function subject to constraints.
Lagrange Multipliers are commonly used for constrained optimization.
What is the first-order condition for necessary optimization?
The gradient must be zero.
This means that the partial derivatives with respect to all variables must equal zero.
What does the Hessian matrix indicate in optimization?
It helps determine whether a point is a local maximum, local minimum, or saddle point.
This involves checking the second derivative for one variable or the determinant for two variables.
