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

In the examples C = {1,2,3,4} and D = {3,4,5}

SymbolMeaningExample{ }Set: a collection of elements{1,2,3,4}A ∪ BUnion: in A or B (or both)C ∪ D = {1,2,3,4,5}A ∩ BIntersection: in both A and BC ∩ D = {3,4}A ⊆ BSubset: A has some (or all) elements of B{3,4,5} ⊆ DA ⊂ BProper Subset: A has some elements of B{3,5} ⊂ DA ⊄ BNot a Subset: A is not a subset of B{1,6} ⊄ CA ⊇ BSuperset: A has same elements as B, or more{1,2,3} ⊇ {1,2,3}A ⊃ BProper Superset: A has B's elements and more{1,2,3,4} ⊃ {1,2,3}A ⊅ BNot a Superset: A is not a superset of B{1,2,6} ⊅ {1,9}A^cComplement: elements not in AD^c = {1,2,6,7}
When  = {1,2,3,4,5,6,7}A − BDifference: in A but not in B{1,2,3,4} − {3,4} =

The Bayes’ theorem describes the probability of an event, based on conditions that Might be related to the event. Bayes’ theorem can also be calculated as the Joint probability divided by the marginal.
Bayes’ theorem deals with conditional probability.

????(????) ∗ ????(????|????) = ????(????) ∗ ????(????|????)

????(????|????)=P(A)P(B|A)/P(B)

????(????|????)=P(Ai)P(B|Ai)/ P(A1)P(B|A1)+ P(A2)P(B|A2)

????(????|????)=P(Ai ∩B) / P(A1∩B)+P(A2∩B)

-Discrete probab distrib(numb of students absent today, numb of books borrowed from library, number of head appering when a cons is tossed tre time)LIMITED

-Conitnous:liter of rain that occur in barcelona, the wheight of all population in spain )UNLIMITED.

Ejercicio Larry

Are indoeendent?

Not because

Joint Cost Allocation Methods

Joint costs are costs of a production process that yields multiple products (co-products). Here's a breakdown of common allocation methods:

Criterion: Market Value

This method allocates joint costs based on the relative sales value of each co-product. Here's how it works:

1. Calculate Total Sales Value:
   • List each co-product (e.g., A, B, C).
   • Determine the sales value per unit (e.g., per kg).
   • Determine the quantity of each co-product produced (e.g., in kg).
   • Multiply the sales value per unit by the quantity to find the total sales value for each co-product.
2. Distribute Total Joint Costs:
   • Calculate the total sales value of all co-products.
   • For each co-product, divide its total sales value by the total sales value of all co-products.

Chapter 2: Predetermined Overhead Rate

Predetermined Overhead Rate = Estimated Total Manufacturing Overhead (MOH) / Estimated Total MOH Driver (e.g., Direct Labor hours, Direct Labor costs, Machine Hours)

Prime Cost = Direct Materials + Direct Manufacturing Labor

Conversion Cost = Direct Manufacturing Labor + Indirect Manufacturing Overhead

Cost Accumulation: Data is collected in an organized way (also known as cost pools).

Cost Assignment: Systematically links an actual cost pool to a distinct cost object (e.g., Tires, engine, labor assigned to car cost).

Activity Base: Examples include kilometers driven in a car, units produced, units sold, machine hours.

Product Cost: Costs tied to creating a product (Direct Materials, Direct Labor, Manufacturing...

Geometry

Angles

Adjacent angles - that have a common vertex and side, but no common interior points

Angle - formed by two rays with a common endpoint called the vertex

Complementary angles - whose measures add up to 90°

Congruent angles - that have the same measure

Supplementary angles - two angles whose measures add up to 180°

Vertical angles - the nonadjacent angles formed by two intersecting lines

Lines

Midpoint - the point that divides a segment into two congruent segments

Parallel lines - in a plane that never meet

Perpendicular lines - that intersect at 90° angles

Transversal - a line that intersects any two or more other lines

Triangles

Triangle Inequality Theorem - the sum of the lengths of any two sides of a triangle is greater than the length...

Small-Scale vs. Large-Scale Risk Aversion

The core idea is to understand the differences between how small and large changes in wealth affect risky gambles.

Diminishing marginal utility (risk aversion) primarily applies to large-scale gambles. This is because the utility function is sufficiently concave over lifetime changes in wealth. This concavity results in a higher utility for taking a certain outcome than for taking a gamble, even if the gamble has a higher expected return.

However, for small-scale gambles, the utility function is locally linear, yielding almost risk-neutral behavior. For wealthy individuals, the utility function is very weakly concave, leading to an asymptotically linear curvature. Thus, diminishing marginal utility cannot...

Discrete Math Concepts

Binary Relations

Cartesian Product

The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.

Example: If A = {0, 1}, the Cartesian product A × A is {(0,0), (0,1), (1,0), (1,1)}.

Properties of Binary Relations

A binary relation R on a set A can have several properties:

1. Reflexive: For all x ∈ A, (x, x) ∈ R.
2. Symmetric: For all x, y ∈ A, if (x, y) ∈ R, then (y, x) ∈ R.
3. Transitive: For all x, y, z ∈ A, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
4. Irreflexive: For all x ∈ A, (x, x) ∉ R.
5. Antisymmetric: For all x, y ∈ A, if (x, y) ∈ R and (y, x) ∈ R, then x = y.

Examples of Relation Properties

Consider a set A = {0, 1}.

• The relation R = A

Practicing material for Quiz #5 – Discrete Probability Distribution SOLUTIONS

1. The manager of a baseball team has determined that the number of walks, x, issued in a game by one of the pitchers is described by the probability distribution given below.

1. This pitcher issues as few as 0 walks and as many as 5 walks in a game.

1. Determine the following probabilities

i. P(x = 2) = 0.15

ii. P(x > 4) = 1 – 0.10 = 0.90

iii. P(x > 5) = 0

iv. P(2 < x < 4) = 0.15+0.45+0.15 = 0.75

1. Calculate the mean for this discrete probability distribution, µ = 2.85 walks

µ = Sxp(x) = 0(0.05) + 1(0.10) + 2(0.15) + 3(0.45) + 4(0.15) + 5(0.10) = 2.85

1. The average, over time, of walks issued in a game by one of the pitchers is 2.

Polygon Interior Angles Theorem

The sum of the measures of interior angles of an n-gon is (n-2)x180

Interior angles of a quadrilateral:

The sum of the measures of the interior angles of a quadrilateral is 360^o

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.

Angles have to measure up to 360. 360/n (n=#of sides)

Theorems

If a quadrilateral is a parallelogram, then its opposite sides are congruent

If a quadrilateral is a parallelogram, then its opposite angles are congruent

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

If a quadrilateral is a parallelogram, then its diagonal bisects each other

If both pairs of opposite sides of a quadrilateral

Axiom

Statement that is considered to be self evident and assumed to be true without any proof or demonstration.

Theorem

Statement that has been proved on the basis of previously established statements, such as other theorems and generally accepted statements such axioms.

Corollary

Statement that follows with little or no proof required from an already proven statement.

Lemma

Mathematical result that is useful in establishing the truth values of some other results.

Trivial proof

The statement on the proof is trivial if we can prove that Q(x) is true for all x in S, then for all x in S, p(x) => q(X) is true regardless of the truth value of p(x).

Vacuous proof

The statement on the proof is vacuous if we can prove that P(x) is true for all x in S, then...