Lecture 1 – Introduction
Mathematics Statistics
- Notation - Population, Sample & Random
- Functions & Equations Variable
- Optimising Functions - Distributions
- Distribution Moments
- Law of Iterated Expectations
- Law of Large Numbers &
Convergence
Mathematics – Notation
→ Intervals & Sets
● We will be working with real numbers (2, – 56.57, 13/28, √2, π)
○ Not real numbers: √-1, ∞
● A set is a collection of real numbers
○ {1, 68}, {2}, {all odd numbers between 5 & 187}
○ Special sets:
■ Set of all real numbers: ℝ
■ Set of all natural numbers: ℕ = {0, 1, 2, …}
● An interval is the set of all real numbers between two fixed endpoints
○ Closed interval includes both endpoints: [a, b]
○ Open interval excludes both endpoints: (a, b)
○ Half-open (or half-closed) interval only includes one endpoint: (a, b] or [a, b)
→ Sigma Notation
● Summation operator (capital sigma: ∑), means ‘add up all of the following elements’
○ Suppose we have a set of four numbers x1, x2, x3 and x4 then:
4
∑ 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4
𝑖=1
○ Subscript i is an index (i = 1 is lower limit, i = 4 is upper limit of sum)
Properties of the Sigma Notation
Commutativity
, Distributivity
for a constant c
Summing a
constant c
Mathematics – Functions & Equations
→ Definition of a function
● A function describes the relation between two or more variables.
○ Functions relate an outcome variable y to input variables x1, x2, …
○ A general function is denoted by f(·)
y = f(x1, x2, …)
○ A univariate function only takes one input variable
y = f(x1)
○ Multivariate functions take multiple input variables
A function y = f(x1, x2, …) relates one outcome value y to each possible set of input variables
{x1, x2, …}
→ Equations
● Equations are mathematical expressions that indicate an equality
○ Simple examples: 1 = 1, 2 + 2 = 4
○ Functions are equations in one or more variables
2
y = 𝑥1
3 2
y = 5𝑥1 − 16𝑥2 + 43
■ We can calculate the outcome (y) by substituting for the input variables 𝑥1
and 𝑥2
■ Given an outcome (y), we can also determine which the set of input
variables that are consistent with this outcome
, ● Linear equation with one unknown
○ a, b, c, d = known constants
○ x = unknown variable
ax + b = cx +d
How to solve for x?
○ Goal is to isolate x on one side of the equality
○ Valid manipulations to achieve this are:
■ Adding (subtracting) the same value to (from) both sides of the equation.
■ Multiplying (dividing) both sides of the equation with (by) the same value
Example 1 Example 2
● System of equations: 2 equations in 2 unknowns
○ {𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝑒𝑖, 𝑓𝑖} i ∈ {1 ,2} = known constants
○ 𝑥1 and 𝑥2 = unknown variables
𝑎1𝑥1 + 𝑏1𝑥2 + 𝑐1 = 𝑑1𝑥1 + 𝑒1𝑥2 + 𝑓1
𝑎2𝑥1 + 𝑏2𝑥2 + 𝑐2 = 𝑑2𝑥1 + 𝑒2𝑥2 + 𝑓2
How to solve for 𝑥1 and 𝑥2?
1. Isolate 𝑥1 in Equation (1) as a function of 𝑥2
2. Substitute this into Equation (2) to find the solution for 𝑥2
3. Substitute the solution from step 2 into step 1, to find the solution for 𝑥1
Example: consider the following system of two linear equations
𝑥1 − 3 = 2 − 𝑥2
2𝑥1 − 𝑥2 = 1
Step 1: isolate one variable (here 𝑥2 is most convenient)
2𝑥1 − 𝑥2 = 1 → + 𝑥2 → – 1 → 𝑥2 = 2𝑥1 − 1
, Step 2: substitute the result into the unused equation to find solution
𝑥1 − 3 = 2 − 𝑥2 | 𝑥2 = 2𝑥1 − 1
𝑥1 − 3 = 2 − [2𝑥1 − 1]
𝑥1 − 3 = 2 − 2𝑥1 + 1
𝑥1 − 3 = 3 − 2𝑥1 | + 2𝑥1
3𝑥1 − 3 = 3 |+3
3𝑥1 = 6 |/3
𝑥1 = 2
Step 3: substitute solution of step 2 into the solution of step 1
𝑥2 = 2𝑥1 − 1 | 𝑥1 = 2
𝑥2 = 2 × 2 − 1 = 3
CHECK: substitute solution into the original equations to check
● Simplifying simple quadratic equations
○ Three simple quadratic equations often show up
2 2
(𝑎 + 𝑏) (𝑎 − 𝑏) (𝑎 + 𝑏) (𝑎 − 𝑏)
● Derivatives of functions
○ The first derivative of a function y = f(x) is denoted:
∂𝑦 ∂𝑓(𝑥)
∂𝑥
= ∂𝑥
= 𝑓'(𝑥) → It describes the change in y when x increases marginally.
○ The second derivative is the derivative of the first derivative:
2
∂𝑦 ∂𝑓'(𝑥)
2 = ∂𝑥
= 𝑓''(𝑥)
∂𝑥
Mathematics Statistics
- Notation - Population, Sample & Random
- Functions & Equations Variable
- Optimising Functions - Distributions
- Distribution Moments
- Law of Iterated Expectations
- Law of Large Numbers &
Convergence
Mathematics – Notation
→ Intervals & Sets
● We will be working with real numbers (2, – 56.57, 13/28, √2, π)
○ Not real numbers: √-1, ∞
● A set is a collection of real numbers
○ {1, 68}, {2}, {all odd numbers between 5 & 187}
○ Special sets:
■ Set of all real numbers: ℝ
■ Set of all natural numbers: ℕ = {0, 1, 2, …}
● An interval is the set of all real numbers between two fixed endpoints
○ Closed interval includes both endpoints: [a, b]
○ Open interval excludes both endpoints: (a, b)
○ Half-open (or half-closed) interval only includes one endpoint: (a, b] or [a, b)
→ Sigma Notation
● Summation operator (capital sigma: ∑), means ‘add up all of the following elements’
○ Suppose we have a set of four numbers x1, x2, x3 and x4 then:
4
∑ 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4
𝑖=1
○ Subscript i is an index (i = 1 is lower limit, i = 4 is upper limit of sum)
Properties of the Sigma Notation
Commutativity
, Distributivity
for a constant c
Summing a
constant c
Mathematics – Functions & Equations
→ Definition of a function
● A function describes the relation between two or more variables.
○ Functions relate an outcome variable y to input variables x1, x2, …
○ A general function is denoted by f(·)
y = f(x1, x2, …)
○ A univariate function only takes one input variable
y = f(x1)
○ Multivariate functions take multiple input variables
A function y = f(x1, x2, …) relates one outcome value y to each possible set of input variables
{x1, x2, …}
→ Equations
● Equations are mathematical expressions that indicate an equality
○ Simple examples: 1 = 1, 2 + 2 = 4
○ Functions are equations in one or more variables
2
y = 𝑥1
3 2
y = 5𝑥1 − 16𝑥2 + 43
■ We can calculate the outcome (y) by substituting for the input variables 𝑥1
and 𝑥2
■ Given an outcome (y), we can also determine which the set of input
variables that are consistent with this outcome
, ● Linear equation with one unknown
○ a, b, c, d = known constants
○ x = unknown variable
ax + b = cx +d
How to solve for x?
○ Goal is to isolate x on one side of the equality
○ Valid manipulations to achieve this are:
■ Adding (subtracting) the same value to (from) both sides of the equation.
■ Multiplying (dividing) both sides of the equation with (by) the same value
Example 1 Example 2
● System of equations: 2 equations in 2 unknowns
○ {𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝑒𝑖, 𝑓𝑖} i ∈ {1 ,2} = known constants
○ 𝑥1 and 𝑥2 = unknown variables
𝑎1𝑥1 + 𝑏1𝑥2 + 𝑐1 = 𝑑1𝑥1 + 𝑒1𝑥2 + 𝑓1
𝑎2𝑥1 + 𝑏2𝑥2 + 𝑐2 = 𝑑2𝑥1 + 𝑒2𝑥2 + 𝑓2
How to solve for 𝑥1 and 𝑥2?
1. Isolate 𝑥1 in Equation (1) as a function of 𝑥2
2. Substitute this into Equation (2) to find the solution for 𝑥2
3. Substitute the solution from step 2 into step 1, to find the solution for 𝑥1
Example: consider the following system of two linear equations
𝑥1 − 3 = 2 − 𝑥2
2𝑥1 − 𝑥2 = 1
Step 1: isolate one variable (here 𝑥2 is most convenient)
2𝑥1 − 𝑥2 = 1 → + 𝑥2 → – 1 → 𝑥2 = 2𝑥1 − 1
, Step 2: substitute the result into the unused equation to find solution
𝑥1 − 3 = 2 − 𝑥2 | 𝑥2 = 2𝑥1 − 1
𝑥1 − 3 = 2 − [2𝑥1 − 1]
𝑥1 − 3 = 2 − 2𝑥1 + 1
𝑥1 − 3 = 3 − 2𝑥1 | + 2𝑥1
3𝑥1 − 3 = 3 |+3
3𝑥1 = 6 |/3
𝑥1 = 2
Step 3: substitute solution of step 2 into the solution of step 1
𝑥2 = 2𝑥1 − 1 | 𝑥1 = 2
𝑥2 = 2 × 2 − 1 = 3
CHECK: substitute solution into the original equations to check
● Simplifying simple quadratic equations
○ Three simple quadratic equations often show up
2 2
(𝑎 + 𝑏) (𝑎 − 𝑏) (𝑎 + 𝑏) (𝑎 − 𝑏)
● Derivatives of functions
○ The first derivative of a function y = f(x) is denoted:
∂𝑦 ∂𝑓(𝑥)
∂𝑥
= ∂𝑥
= 𝑓'(𝑥) → It describes the change in y when x increases marginally.
○ The second derivative is the derivative of the first derivative:
2
∂𝑦 ∂𝑓'(𝑥)
2 = ∂𝑥
= 𝑓''(𝑥)
∂𝑥
