Notes, summaries, assignments, exams, and exercises for Mathematics for Baccalaureate

1. Present Value of an Investment

15 Years Case

Present Value of the Investment = $4,900 [ (1/0.08) − (1/0.08(1 + 0.08)^15)] = $41,941.45

2. Effective Annual Rate (EAR)

EAR = (1 + (APR/m))^m − 1 If m becomes infinite, EAR = e^APR − 1

7% APR and quarterly compounding: EAR = (1 + (0.07/4))^4 − 1 = 7.19%

16% APR and monthly compounding: EAR = (1 + (0.16/12))^12 − 1 = 17.23%

11% APR and daily compounding: EAR = (1 + (0.11/365))^365 − 1 = 11.63%

3. Value of a Perpetual Stream of Payments

Present Value of the Perpetuity = (Payment/Interest Rate) Payment = $2,500, Interest Rate = 0.061

PV of the perpetuity at date t=14 is $2,500/0.061 = $40,983.61. Discounting it back to date t=7, we should have PV7 = $40,983.61/(1+0.061)^7 = $27,077.12.

4. Bond

Relative Positions in 3D Space

Position of Three Planes

The relative position of three planes is determined by analyzing the rank of the system of equations.

• Intersect at one point: The rank of the coefficient matrix and the augmented matrix is 3.
• Intersect in a line: The rank of both matrices is 2.
• Parallel or Coincident: If the ranks are 1 or 2, the planes can be parallel, coincident, or form a prismatic surface.

Position of Two Lines

Given two lines with direction vectors v₁, v₂ and points P₁, P₂:

• Intersecting Lines: The lines are coplanar. The determinant of the matrix formed by vectors v₁, v₂, and P₁P₂ is zero. To find the intersection point, solve the system of their parametric equations.
• Parallel or Coincident Lines: The direction

Key Statistical Concepts

Understanding Percentiles

The Xth percentile means X% of the data must fall strictly below it. The percentile of X can be calculated using the formula: (# observations (N - 1) / 2 / N * 100%).

Variance: Population vs. Sample

• The sample variance is the sum of the squared deviations from the mean divided by the number of measurements minus one.
• The population variance is the sum of the squared deviations from the mean divided by the number of measurements.

The Empirical Rule

Also known as the 68-95-99.7 rule, the Empirical Rule states that for a normal distribution:

• Approximately 68% of the measurements will fall within one standard deviation of the mean.
• Approximately 95% of the measurements will fall within two standard deviations

Are X and Y independent random variables?

Solution: First we calculate the marginal pdfs

Note that fX,Y (x, y) 6= fX(x) × fY (y). So they are not independent.

What is the conditional distribution of X, given Y = .50? What is the probability that less than 50% of the initiative 1 surveys are returned, given that 50% of the initiative 2 surveys are returned?

(c) Calculate E(X/Y = y). What is the expected proportion of initiative 1 surveys returned, given that 50% of the initiative 2 surveys are returned?

Three prisoners are informed by their jailer that one of them has been 1/3 1/2

First we label the prisoners as P1, P2 and P3. Suppose P1 asks the jailer to tell him privately whether P2 or P3 will be set free. Now we define the following events.

Ai...

The Production Volume Variance

Represents the difference between budgeted (incurred) and actual (allocated) fixed manufacturing costs.

When a greater proportion of costs are fixed costs, then

when demand is low the risk of loss is high

FALSE?

Gross Margin will always be greater than contribution margin.

Given a constant contribution margin per unit and constant fixed costs, the period-to-period change in operating income under variable costing is driven solely by:

changes in the quantity of units actually sold

In CVP analysis, the amount of units to be sold for the company to break even:

Increases when the contribution margin per unit decreases, all other things being equal

FALSE?

Nonmanufacturing costs are expensed in the future under variable costing....

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Handshaking Lemma

In any undirected graph, the sum of the degrees of all vertices is twice the number of edges.

Formula: Σdeg(v) = 2|E|

Euler's Formula for Planar Graphs

For planar graphs, the relationship between vertices (V), edges (E), and regions (R) is defined as:

V - E + R = 2

Sum of Degrees and Odd Vertices

The sum of the degrees of all vertices in any graph is always even because each edge contributes 2 to the total sum. Furthermore, the number of vertices with an odd degree must always be even.

Graphs with No Odd Degree Vertices

If all vertices in a graph have an even degree, the graph is Eulerian, meaning it contains an Eulerian circuit.

Complete Graphs

A complete graph with n vertices, denoted as K_n, has an edge between every pair of distinct...

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1) What is an Inverse Matrix and How Do You Calculate It?

A matrix A^-1 is called the inverse matrix of a matrix A (nxn) if AxA^-1= A^-1xA=E (where E is the identity unit matrix).

We calculate it by performing row operations on the augmented matrix (A | I) to transform it into (I | B). If this reduction is possible, then B=A^-1, which is the inverse matrix of A.

2) Define the Model of a Game

Games can be modeled in various forms:

• Tree Form Model (Game Tree): Represents the game as a sequence of decisions (moves) made by players.
• Normal Form Model: Represents the game using:
  • List of players
  • List of strategy spaces for each player
  • List of payoff functions (decision matrix) defining outcomes for each combination of strategies.
• Characteristic Function Form: