Statistics

Set

defined by brackets or intervals

• Intervals: defines the set of all the numbers inclusive

closed interval [-2, 3] all numbers from -2 to 3 inclusive

open interval (-2, 3) all numbers between but -2 and 3

unbounded interval (-∞,∞)

• Containment(∈) and shared properties

~ | ~ ~ such that ~

N_even = {2n∈N|n∈N} the set of numbers formed by doubling all the natural numbers

[a,b] = {x∈R: a≤x≤b}

• Subsets(⊂): a collection of elements from within a particular set

Z⊂R

• Empty set(∅): the only set that contains no elements at all
• Universal set(U): more general set which contains all
• Union(∪): combine all the elements from the original sets into a single set
• Intersection(∩): elements are common to both of the two sets
• Complement of a set S(¬S): containing elements which are not in S

S ∩ ¬S = ∅

• Compound relations: combination of intersections and/or unions
• Venn diagram: picturing operations between sets

• Sample space: a set consisting of all possible results of a trial or experiment or all events associated with a particular process. An individual outcome or result is also called an event.
• Propositions: mathematical logic dealing with basic statements

can be true, false, or indeterminate

• Compounds statements: all relations between two+ propositions
  • Implication(p⇒q) if p, then q (p⇔q) q if and only p
  • Conjunction (∧) = and
  • Disjunction (∨) = or, possibly both
  • Exclusive disjunction = or but not both

• Truth tables
• Contradiction p and ¬p cannot be true at the same time
  • Tautology: proposition always true

• Implication: p⇒q
  • Converse: q⇒p
  • Inverse: ¬p⇒¬q
  • Contrapositive: ¬q⇒¬p

• Containment 3∈Z, -3∉Z
• Subset Z⊂N A⊆Z
• Empty set {}
• Universal set U
• Union A ∪ B
• Intersection A ∩ B
• Complement Q(rational numbers) ∪ Q' = R(real numbers)

• Sample space: a set consisting of all possible results of a trial or experiment, or all events associated with a particular process
• Event: an individual outcome or result

Probability

likelihood of an occurrence

P(¬S)+P(S) = 1

• Monte-Carlo method

Estimate of unknown area = #points in unknown area/ total #points generated x known area

• Tree diagrams

• Types of events
• Combined events P(A∪B) = P(A) + P(B) - P(A∩B)
• Mutually exclusive events P(A∪B) = P(A) + P(B)
• Independent events P(A∩B) = P(A) x P(B)
• Conditional probability P(A|B) = P(A∩B)/P(B)

• Heads and Tails
• Dice

• Venn diagram
• Tree diagram

• Combined events P(A∪B) = P(A) + P(B) - P(A∩B)
• Mutually exclusive events P(A∪B) = P(A) + P(B)
• Independent events P(A∪B) = P(A) x P(B)
• Conditional probability P(A|B) = P(A∩B)/P(B)

Discrete Mathematics

• Proposition: basic statement dealing with a mathematical logic
  • Conjunction ∧ (and)
  • Disjunction ∨ (or)
  • Exclusive disjunction (or but not both)
• Implication ⇒
  • Equivalence ⇔

• Tautology P∨¬P
• Contradiction P∧¬P

• Implication p⇒q
• Converse q⇒p
• Inverse ¬p⇒¬q
• Contrapositive ¬q⇒¬p

Statistics

• Discrete data counted
• Continuous data measured

• Frequency tables
  • Mid-interval values
  • Upper & lower boundaries
• Frequency polygons only for continuous data
• Frequency histograms
• Stem and leaf diagrams

• Cumulative frequency
• Percentiles and quartiles
• Whisker plots
• Outliers

• Measures of central tendency
  • Mode
  • Median
  • Mean

• Measures of dispersion
  • Range
  • Interquartile range = Q₃ - Q₁
  • Standard deviation

• Correlation

• Linear regression

    (y-ȳ) = S_xy (x-¯x) / (S_x)²

• ℵ² test for independence
• ρ-value

• Weighted average

• Geometric mean

• Harmonic mean

• measurement scales(NOIR) - Nominal(no context; weakest), Ordinal(accoding to characteristics), Interval(special meaning to difference to numberical values), Ratio(scale amounts; strongest)

• Correlation: degree of linear dependence between the variables
  • 1 in the case of an increasing linear relationship,
  • −1 in the case of a decreasing linear relationship,
  • some value in between in all other cases
  • the closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

• Bayes formula

• Permutation: order matters nPr = n! / (n-r)!
• Combination: order does not matter nCr = n! / r! (n-r)!

• Mean Absolute Deviation

• Standard deviation

• Chebyshev’s inequality : no more than 1/k2 of the values are more than k standard deviations away from the mean.

• Sharpe ratio: measure of the excess return (or Risk Premium) per unit of risk in an investment asset

• Roy's Safety-First criterion

P(Ra < Rm) = [E(R_P)-R_L]/σ_P

• Skewness
  • measure of the asymmetry
  • Positive skew: mode < median < mean
  • negative skew: mode > median > mean

• Kurtosis
  • measure of the "peakedness"; can be either leptokurtic or platykurtic

Probability Concepts

• Probability distribution: of all possible outcomes for a random variable
  • discrete distribution: finite number of possible outcomes
  • continuous distribution: infinite number of possible outcomes
  • Probability function p(x): probability that a discrete random variable will take on the value x
  • Probability density function f(x): probability a continuous random variable will take on a value within a range
  • Cumulative distribution function F(x): probability a random variable will be less than or equal to a given value
  • Binomial random variable: probability of exactly x successes in n trials

• Confidence interval: a range of values around an expected outcome within which we expect the actual outcome to occur some specified percent of the time
  • 90% confidence interval = Χ ± 1.65σ
  • 95% confidence interval = Χ ± 1.95σ
  • 99% confidence interval = Χ ± 2.58σ

• Degrees of freedom
  • sufficiently high df is approximately normal
  • higher degrees of freedom, thiner tails

• Standard normal distribution: μ=0 σ=1
  • z-value standardization

              where   plug-in z-value to get F(z) from z-table

• Covariance

• Coefficient of Variation

    C_V = σ / μ

• Monte Carlo Simulation: to estimate distribution of derivatives prices or of Net Present Values

• Continuous compounding = ln(1+HPR)

Sampling and Estimation

• Sampling: to make inferences about the parameters of a population
  • time-series data- gathered from each time periods
  • cross-sectional data- data from a single time period

• Stratified sampling: random picks within subgroups

• Central limit theorem
  • sample mean for large sample sizes will be distributed normally
  • as sample size increases, becomes more accurate in respect to population data
  • holds for n > 30

• Test statistic: difference between population sample and hypothesized value

• Z-test

• Standard error

• Student's t-test
• used when sample size is small or variance unknown

• Chi square test

used when H0: σ² = σ₀<super>2</super>

• Confidence interval (level of significance:probability of rejecting true H₀)
  • 68% of observations fall in ±1σ
  • 95% of observations fall in ±1.96σ
  • 99% of observations fall in ±3σ

• Types of Bias
  • data-mining~ : repeatedly doing tests on same data sample
  • sample selection~ : sample not really random
  • survivorship~ : sampling only surviving firms
  • look-ahead~ : using information not available at the time to construct sample
  • time-period~ : relationship exists only during the time period of sample data

Hypothesis Testing

• • H₀: hypothesis set up to be nullified or refuted in order to support an alternate hypothesis.
  • H₁: alternative hypothesis
  • Type I error(Significance level): rejecting a null hypothesis when it is actually true; decreases as confidence interval(tradeoff) increases
  • Type II error(1 - Power of test): failing to reject a null hypothesis when the alternative hypothesis is the true; increases as confidence interval increases

• Volatility estimation
  • unbiased~ : has an expected value equal to the true value of the population parameter
  • consistent~ : more accurate the greater the sample size
  • efficient~ : has the sampling distribution that is less than that of any other unbiased estimator

• Statistical significance omits transaction costs, taxes, risk factor from economical significance

Geometry and Trigonometry

Cartesian Geometry

Brief overview of basics

• Midpoint 
  

    

• Pythagorean theorem

    d = 

• Gradient(slope)

m = 

• Parallel lines

two lines are parallel if and only if they have the same gradient

• Perpendicular lines: negative reciprocal

negative reciprocal of m₁ = 1/m₁

• Collinear points: points which lie on the same straight line
• Axis intercepts

y-intercept intersection of a line with the y-axis when x=0

x-intercept intersection of a line with the x-axis when y=0

• Point of intersection

no point of intersection => different (likely parallel) lines

1 point of intersection => different gradients

infinite number of intersection points => coincident lines

Right-angled Trigonometry

• Trigonometric ratios

• Finding the sides and the angles of a right-angled triangle
  • Isosceles triangle
  • Any triangle
  • Rectangle
  • Rhombus
  • Trapezium
  • Parallelogram
  • Circle

• The sine rule

• The cosine rule

• Area of a triangle

Geometry of Solids

Solid	Area of curved face	Volume
Cone		V = Area of a base x height / 3
Sphere
Cylinder

Calculus

• Gradient of a curve

• Finding the gradient of a curve at a general point P

• Derivative of polynomials

when y = x^n , where n is a real number, 

when y = a * x^n, where n is a real number, 

when  , where u and v are functions of x, 

• Negative powers

• Second derivative ∂²y/∂x²

• Tangent equation

• Gradient of the tangent line

• Increasing and decreasing functions

• Zero gradient

points where f'(x) = 0

• Local maxima and minima

• Stationary point

• Point of inflexion

Quadratic functions

y = ax² + bx + c

• parabola shape with axis of symmetry

• Completed square form F(x)=a(x-r)² + s

axis of symmetry at r

check that f(r-h) = f(r+h)

Factorising techniques

• 3 quick special cases
  1. No constant term x(x-r)
  2. Difference of two squares (x-r)(x+r)
  3. Sum of two squares (x-r)(x-r)

never factorize

• General case hit-and-miss process
  1. Find factors for a and c
  2. Test for the case matching b

• Quadratic formula

        

• Transformation into quadratic equation

• Point of intersection

• Fatorising a polynomial
  1. Factor theorem For polynomial p(x), if p(t) = 0, then (x-t) is a factor of p(x).
  2. Extended factor theorem If p(t/s) = 0, then (sx-t) is a factor of p(x).
  3. Method of equating coefficients If ax²+bx+c = lx²+mx+n, then a=l, b=m, c=n.

• The remainder theorem

If a polynomial p(x) is divided by a linear polynomial sx-t, the remainder is p(t/s).

Exponentiation

• Power functions f(x) = x^a

• Negative integer powers

• Fractional indices

• Exponential functions f(x) = a^x

Investigating shapes of graphs

• Exponential functions

f(x) = ka^λx + C

• Stationary points: points of a graph at which the gradient is 0

• Local maxima and minima
  • f'>0 increasing function
  • f'<0 decreasing function
  1. Find an expression for f'(x)
  2. List the values of x for which f'(x) = 0
  3. Find the sign of f'(x) in intervals to the left and to the right of that value
  4. If these signs are - and + respectively, => minimum point. If + and -, => maximum point
  5. For each value of x which gives a minimum or maximum, calculate y = f(x)

• Application to roots of equations

Second derivatives

• examines the way in which the graph is bending
  • f" > 0 => concave up
  • f" < 0 => concave down
  • f" = 0 ?<=>? Point of inflexion f"

• Local maxima and minima
  1. Find f'(x)
  2. List the values of x for which f'(x)=0
  3. Find f"(x)
  4. For each x with f'(x)=0, find the sign of f"(x). If the sign is + => minimum point; if - => maximum
  5. For each value of x which gives a minimum or maximum, calculate y = f(x)

Differentiation

• Gradient:
• Differentiation: process of finding the gradient formula for a curve
• Derivative: the gradient formula for a curve

• Differentiation rule

If f(x) = xⁿ, where n is a positive integer, then f'(x) = nx^n-1

The derivative of af(x) + bg(x), where a and b are constants, is af'(x) + bg'(x)

proof f(x)=ax²+bx+c

Integration

reverse process of differentiation

• Finding a function from its derivative

ex.> f'(x) = 2x + x²

• Indefinite integral ∫f(x) dx
• Arbitrary constant

ex.> f'(x) = 6x² - 5x; f(x) passes through (2,3)

• Application to kinematics

ex.> 28.3.1 For the first few seconds of a race a horse's acceleration, a m/s², is modelled by the equation a = 6 - 1.2t, where t is the time in seconds from a standing start. Find an expression for the distance it covers in the first t seconds. Hence find the horse's acceleration 5 seconds after the start, how fast it is then moving and how far it has run.

ex.> 28.3.2 A train is traveling on a straight track at 48 m/s when the driver sees an amber light ahead. He applies the brakes for a period of 30 seconds, producing a deceleration of 1/125 * t(30-t) m/s², where t is the time in seconds after the brakes are applied. Find how fast the train is moving after 30 seconds, and how far it has traveled in that time.

• The area under a graph

to find the area under the graph f(x) from x=a to x=b:

1. Find the simplest integral of f(x), call it I(x)
2. Work out I(a) and I(b)
3. Area = I(b) - I(a)

ex. 29.1.2 Find the area between the graph of y=2x - x² and the x-axis.

• Definite integrals

ex.> 29.3.1 An aircraft accelerates along the runway as it takes off. After t seconds it has traveled s meters and has a speed of v m/s. For 10<t<20 the motion is modeled by the equation v = 100 - 4000/t². Find how far the aircraft travels in this time.

• Properties

ex.> Find the area between y=x² and the chord joining the points (-1,1) and (2,4) on the curve

ex.> 29.7.2 Show that the graphs of y=f(x) and y=g(x), where f(x)=x³ - x² - 6x + 8 and g(x) = x³ + 2x² - 1, intersect at two points, and find the area of the region enclosed between the two curves.

ex.> 29.7.3 y=3 + 2x - x² Calculate the areas enclosed

Exponentials & Logarithms

• Exponential function f(x) = b^x

where b is a positive real number and base b≠1

• Logarithm to base b of y

if x is a real number and y is a positive real number

b^x = y is equivalent to x = log_by

examples

• Properties(p.439-441)
  • Multiplication rule
  • Division rule
  • Power rule
  • the nth root rule

• Special bases
• log base 10
• natural log

Extension

consider a function in the following format: 

• Differentiation

• Integration

• Sum rule

     if y = u+v

• Product rule

     if y = uv

• Quotient rule

 if  

Exponentials and Logarithms

• Exponential function ,  

• e = 2.718271828...

for which f'(0) = 1

• for any positive base b,

• Natural logarithm

, 

• Natural log differentiation

• Reciprocal integral

• Irrational indices

Volumes of revolution

• Rotation about the x-axis

• Rotation about the y-axis

• Rotating regions between curves

Chain Rule

Circular functions

• Differentiation

• Applications

Integrating products

• Integration by parts

• Arbitrary constant

Integration by parts

Differential Equations

the rate of change of one variable with respect to the other

• A rodent has mass 30 grams at birth. It reaches maturity in 3 months. The rate of growth is modelled by the differential equation , where m grams is the mass of the rodent t months after birth. Find the mass of the rodent when fully grown.

• A botanist makes a hypothesis that the rate of growth of hothouse plants is proportional to the amount of daylight they receive. If t is the time in years after the shortest day of the year, the length of effective daylight is given by the formula 12 - 4 cos 2π t hours. On the shortest day the height of one plant is measured to be 123.0 cm; 55 days later the height is 128.0 cm. What will its height be on the longest day off the year six months later?

• Gardeners are concerned about the spread of a species of beetle. All the specimens detected so far lie within a circular region of radius 25 kilometers, and it is suggested that the increase of the radius r kilometers might be modelled by a differential equation , where t denotes the time in months. What does this model predict for the radius of the region colonised by the beetle after t months?

• A hot air balloon can reach a maximum height of 1.25 km, and the rate at which it gains height decreases as it climbs, according to the formula , where h is the height in km and t is the time in hours after lift-off. How long does the balloon take to reach a height of 1 kim?

• When a ball is dropped from the roof of a tall building, the greatest speed that it can reach(terminal speed) is u. One model for its speed v when it has fallen a distance x is given by the differential equation , where c is a positive constant. Find an expression for v in terms of x.

• A steel ball is heated to a temperature of 700 degrees Celsius and dropped into a drum of powdered ice. The temperature falls to 500 degrees in 30 seconds. Two models are suggested for the temperature, T degrees, after t seconds:
  • the rate of cooling is proportional to T
  • the rate of cooling is proportional to T^1.2

It is found that it takes a further 3 minutes for the temperature to fall from 500 to 100 degrees. Which model fits this information better?

Integration by substitution

• Direct substitution

• Definite integrals

If x=s(u), then 

 where g(u) = f(s(u)), and p = s^-1(a), q = s^-1(b)

• Reverse substitution

If u=r(x), and if g(r(x)) = f(x), then   is equal to , with u replaced by r(x).

Matrix

• elements: individual numbers in the matrix
• order p x q
• rows p
• columns q

• Scalar addition

• Scalar multiplication

• Matrix multiplication

(row x column)

• Identity matrix

IX = XI = X

• Zero matrix

A+O = O+A = A

For a function 

• Adjugate

• Determinant

• Inverse matrix

AC = CA = I

AA^-1 = A^-1A = I

Vectors

• Displacements in the plane in 3-D

• Components of a vector

• Scalar product
  • Perpendicular vectors v . w = 0
  • Parallel vectors v . w = |v||w|

• Magnitude

• Unit vectors

• Position vectors