These a are supplementary notes for week 3. They elaborate
the material

presented in lecture and provide practice problems. Please also read Hagle

chapters 1 and 2.

**1 Common Math Notation and Review
**

Notation for calculus and linear algebra is presented in week 10. This section

presents general mathematical and pre-calculus notation and terms.

• Axiom: Something we take as given (we do not prove that it is true)

• Constant: A fixed number that does not change

• Denominator: Top part of a fraction

• Numerator: The lower part of a fraction

• Proposition: A rule or formula that we can prove to be true

• Variable: In math, this is something that is unknown or something that

can take finitely or infinitely many values. In statistics it is something

that varies accross observations.

1.2.1 Parantheses and Brackets

The curvy brackets {} are used to indicate sets. For example, S={1,2,3} means

that the set S contains the values 1, 2, and 3.

The square brackets [ ] and the parentheses () are used to indicate open

and closed intervals. An open interval does not include its endpoints whereas

a closed interval does include its endpoints. Open intervals use parentheses

and closed intervals use square brackets. For example, the interval (1,5) is the

interval from 1 to 5, exclusive – this means that 1 and 5 are not included in the

interval. The interval [1,5] is the interval from 1 to 5, inclusive – 1 and 5 are

included in the interval. (1,5] means that 5 is included but 1 is not.

The parentheses are also used to represent points. It should be clear from

context whether (a,b) refers to the open interval from a to b or to the point

(a,b).

Parentheses can also represent multiplication: (2)(4) = 2
* 4 = 8. Paren-

theses indicate that some operation (such as multiplication) is done to the

entire expression within the parentheses as a unit. For example, it is use-

ful when multiplying things with more than one component (polynomials):

(1 + x)(2 + 3x) = 2 + 5x + 3x^{2}.

**1.2.2 Infinity
**

Infinity is represented by ∞ and negative infinity (infinitely small; smaller than

zero) is represented by -∞.

The inverse (reciprocal) of a number n is 1/n. It is often expressed as n

is a particular example of an exponent (see below for more about exponents).

1. How do you express the closed interval from negative infinity to two?

2. What is the invserse of ?

Sometimes we need to express a series of numbers (constants) or variables. If

there are infinite or very many items in the list, we often abbreviate it with

ellipses (...) and subscripts. For example:

• 1,2,3,..., 50 represents the integers from 1 to 50. The ellipsis (...) indicates

that numbers are left out; they are implicit. We can infer that the missing

values are integers by extrapolating from the values shown.

• represents a series of 50 values. For example, X might

be a variable measuring the number of sexual partners for each of 50

observations.

• represents a list of n things; n may be variable or unknown.

This is often used to present a general formula that can be applied to any

number of values.

Similar notation is used to represent mathematical operations on a series of

numbers or variables:

• 1 + 2 + ... + 50 indicates that the integers from 1 to 50 are all added

together.

• 1+2+...+n indicates that the integers from 1 to n are all added together;

n is variable or unknown.

• 1 * 2 * 3 * ... * n indicates that the integers from 1
to n are all multiplied

together. (1)(2)...(n) is an equivalent notation.

• indicates that the n variables are squared
and multi-

plied together.

**Exercises: Lists
**

1. What is 2(1)+2(2)+...+2(5)? Hint: There are two things left out where

the ellipsis is.

1.4.1 Sums

The simplist way to express a sum is with a plus sign (+). For example, we

can write 1 + 1 = 2. However, when we are summing a series of values, we

sometimes use the greek letter capital sigma (∑) because it allows us to be

much more concise. For example:

This means that we need to sum the series of numbers
represented by from

i = 1 to i = n: . For
example, let X be a variable representing

the number of children in a family. Say we have data on three families:
,

and . We want to know the total number of
children in our data:

When the number of observations (n) is very large, using
the sigma notation is

a much more concise way to express the sum of a series of values. It also allows

us to write general equations for an unspecified n. For example, it is an easy

way to express the formula for the mean of a variable:

The sigma notation is a very concise instruction to add up
all the values for the

n observations and divide by the sample size (n).

**1.4.2 Multiplication**

For simple equations, we represent multiplication by 2*2 = 4 or (2)(2) = 4.

When we need to multiply a list of numbers, we use the number capital pi()

to represent the product accross some series. For example:

This indicates that we multiply the n values together.

**1.4.3 Subtraction and Division
**

Subtraction is addition of a negative number (2 - 1 = 2 + (-1)) and division is

multiplication by an inverse . Therefore, subtraction can be repre-

sented by the ∑ notation and division can be represented by the notation.

1. What is ?

2. What is ?

3. What is ?

Convergence means that a sequence or function approaches some limit. Having

a limit means that the function or sequence gets closer and closer to some value;

this value is the limit. For example:

As x → 1, → 0. The
arrow means “converges to” or “approaches.” Here’s

another example:

This function converges to the constant e. So does this sequence:

The exclamation point is the factorial (See next section,
1.6).

**1.6 The Factorial**

The factorial is indicated with an exclamation point (!). The expression n! is

read “n-factorial.” For some number n, n! equals n(n − 1)(n − 2)...(3)(2)(1). 0!

equals 1 by definition. For example:

It is easy to calculate factorials using Excel (or another
computer program).

For example, in Excel the foruma “=FACT(n)” gives n!.

The factorial is useful in probability because n! gives you the number of ways

to order n things. Why does this work? There are n things to select for the
first

spot, there are n − 1 things to choose for the second spot, n − 2 things for the

third spot... These selections are independent events so we can multiply their

probabilities: n(n − 1)(n − 2)...

**Exercises: The Factorial
**

1. You’re having a three scoop ice cream cone with three different flavors.

How many ways could you make this cone (eg, how many ways to order

the flavors as first, second, and third scoops)?

2. You also have to choose a cone. There are four types of cones. How many

different combinations of cones and ordered scoops of ice cream are there?

3. You’re organizing your books. You have 3 methods books and 4 theory

books. You want to put all the methods books together and all the theory

books together. All 7 books will be on the same shelf. How many orders

are possible?