Mathematics for Statisticians lecture notes

Mathematics for Statisticians

Lecture 1

A set is a collection of numbers, either finitely many or infinitely many. Real numbers are all
non-complex numbers, so numbers that do not involve i. The real numbers’ set has a symbol
ℝ.
A special kind of sets are the intervals, which are all the real numbers between 2 numbers, i.e.
every number between a and b. When it comes to intervals, we choose if we either want to
include the endpoints or not:
1. An open or round bracket means non inclusion
a. (0, 1) means all numbers such that 0 < 𝑥 < 1, so not 0 and 1
2. A closed square bracket means inclusion
a. [0, 1] means all numbers such that 0 ≤ 𝑥 ≤ 1
b. (0, 1] means all numbers such that 0 < 𝑥 ≤ 1

Exercise
Write in notation all numbers between − π and 10, excluding the former and including the
latter.

(− π, 10]

We use ± ∞ to indicate positive or negative infinity. For example, (5, ∞) is any 𝑥 > 5. if infinity
is the endpoint, one must use an open bracket.

Another set notation is where all elements are written within curly brackets. For example,
{1, 2, 3}. another example, we can write [0, 1] as {𝑥 ∈ ℝ: 0 ≤ 𝑥 ≤ 1}.
The symbol ∈ means belonging to a set, so 𝑥 ∈ ℝ means all numbers in ℝ. The symbol : means
“such that”.

Exercise
Use set notation to write the intervals [− 20, π] and (− ∞, 4).

{𝑥 ∈ ℝ: − 20 ≤ 𝑥 ≤ π}
{𝑥 ∈ ℝ: 𝑥 < 4}

A function is a map between two sets that assigns a unique output to each set. The domain
𝐷(𝑓) of a function is the set of all valid inputs and the range 𝑅(𝑓) is the set of all possible
outputs. The notation for functions is:
2
1. 𝑓(𝑥) = 𝑥 , 𝐷(𝑓) = ℝ, 𝑅(𝑓) = [0, ∞)
2
2. 𝑓: 𝑥 → 𝑥 , 𝐷(𝑓) = ℝ, 𝑅(𝑓) = [0, ∞)
Domains can differ. For example, 𝑓(𝑥) = 𝑥 has as domain 𝐷(𝑥) = {𝑥 ≥ 0}.

,Let 𝑓(𝑥) and 𝑔(𝑥) be two functions such that 𝐷(𝑔) ⊆ 𝑅(𝑓), the composition of 𝑓(𝑥) and 𝑔(𝑥)
is the function
(𝑓 ◦ 𝑔)(𝑥) or 𝑓(𝑔(𝑥)).

Exercise
2
If 𝑓(𝑥) = 𝑥 + 1 and 𝑔(𝑥) = 𝑥 − 2, find (𝑓 ◦ 𝑔)(𝑥) and (𝑔 ◦ 𝑓)(𝑥).

2 2
(𝑓 ◦ 𝑔)(𝑥) = (𝑥 − 2) + 1 = 𝑥 + 4𝑥 + 5
2 2
(𝑔 ◦ 𝑓)(𝑥) = 𝑥 + 1 − 2 = 𝑥 − 1

If 𝑓(𝑥) and 𝑔(𝑥) are two functions such that (𝑓 ◦ 𝑔)(𝑥) = 𝑥 = (𝑔 ◦ 𝑓)(𝑥) then 𝑓(𝑥) and 𝑔(𝑥)
are inverse functions and are noted by
−1
𝑔(𝑥) = 𝑓(𝑥) .

Exercise
3 3
Given 𝑓(𝑥) = 𝑥 and 𝑔(𝑥) = 𝑥, calculate (𝑓 ◦ 𝑔)(𝑥) and (𝑔 ◦ 𝑓)(𝑥).

3
3
(𝑓 ◦ 𝑔)(𝑥) = 𝑥 = 𝑥
3 3
(𝑔 ◦ 𝑓)(𝑥) = 𝑥 =𝑥

2
Not all functions have an inverse. For example, the inverse of 𝑓(𝑥) = 𝑥 would be 2, but the
latter is not a function, because the same input can have two outputs. Visually, if, by tracing a
horizontal line on the function’s graph, the line meets the graph in more than one point at any
point, that function is not invertible.

A polynomial of degree 𝑛 is a function in the form
𝑛 𝑛−1
𝑓(𝑥) = 𝑎𝑛𝑥 + 𝑎𝑛−1𝑥 +... + 𝑎1𝑥 + 𝑎0
where 𝑎𝑖 ∈ ℝ and 𝑎𝑛 ≠ 0. A polynomial of degree 1 is a line. A polynomial of degree 2 is a
parabola. The roots or zeros of a function 𝑓(𝑥) are the values for which 𝑓(𝑥) = 0.

Exercise
2
Find the zeroes of 𝑓(𝑥) = 𝑥 + 𝑥 − 6.

−1± 1−4(1)(−6)
2
= 2, − 3


Let 𝑏 > 0, 𝑏 ≠ 1 be a real number. The exponential function is of the form
𝑥
𝑓(𝑥) = 𝑏 .
A few characteristics:
0
1. 𝑏 = 1 ∀𝑏 > 0

, 𝑥
2. 𝑏 ≠ 0 ∀𝑥 ∈ ℝ
𝑥
3. 𝑏 > 0 ∀𝑥 ∈ ℝ
4. 𝑓(𝑥) → ∞ as 𝑥 → ∞ and 𝑓(𝑥) → 0 as 𝑥 → − ∞, if 𝑏 > 1
5. 𝑓(𝑥) → 0 as 𝑥 → ∞ and 𝑓(𝑥) → − ∞ as 𝑥 → − ∞, if 0 < 𝑏 < 1
𝑥
Within the exponentials, we care most about 𝑏 = 𝑒 ≃ 2. 718. The function 𝑒 is called the
(natural) exponential function.

Exercise
𝑥 2 −3𝑥
Simplify (𝑒 ) 𝑒 .

𝑥 2 −3𝑥 2𝑥 −3𝑥 −𝑥 1
(𝑒 ) 𝑒 =𝑒 𝑒 =𝑒 = 𝑥
𝑒


𝑥
The logarithm is the inverse function of 𝑏 is the logarithm with base 𝑏 written
𝑙𝑜𝑔𝑏(𝑥).
If 𝑏 = 𝑒, we write 𝑙𝑛(𝑥), which means natural logarithm. For 𝑏 > 1, 𝑙𝑜𝑔𝑏(𝑥) is always
increasing, and only defined if 𝑥 > 0.
As exponential and logarithm are inverse functions, we have the following properties:
𝑥
1. 𝑙𝑜𝑔𝑏(𝑏 ) = 𝑥
𝑙𝑜𝑔𝑏(𝑥)
2. 𝑏 =𝑥

Exercise
Let 𝑙𝑛(𝑥) = 2. Find x.

𝑙𝑛(𝑥) 2 𝑙𝑛(𝑥) 2
𝑒 = 𝑒 and 𝑒 = 𝑥 which means that 𝑥 = 𝑒 .

Properties of the logarithm:
1. 𝑙𝑜𝑔𝑏(1) = 0
2. 𝑙𝑜𝑔𝑏(𝑏) = 1
𝑟
3. 𝑙𝑜𝑔𝑏(𝑥 ) = 𝑟 × 𝑙𝑜𝑔𝑏(𝑥)
4. 𝑙𝑜𝑔𝑏(𝑥𝑦) = 𝑙𝑜𝑔𝑏(𝑥) + 𝑙𝑜𝑔𝑏(𝑦)
𝑥
5. 𝑙𝑜𝑔𝑏( 𝑦 ) = 𝑙𝑜𝑔𝑏(𝑥) − 𝑙𝑜𝑔𝑏(𝑦)
𝑙𝑜𝑔𝑎(𝑥)
6. 𝑙𝑜𝑔𝑏(𝑥) = 𝑙𝑜𝑔𝑎(𝑏)


𝑛
For sums such that 𝑥1 + 𝑥2 +... + 𝑥𝑛, the notation for sums is ∑ 𝑥.
𝑛=1


Exercise

, Write 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 in notation.

10
∑ 𝑥
𝑛=1


Lecture 2

We say a number 𝐿 is the limit of a function 𝑓(𝑥) as 𝑥 approaches a value 𝑎, if 𝑓(𝑥) gets
arbitrarily close to 𝐿 as 𝑥 approaches 𝑎 from either side. The notation is:
lim 𝑓(𝑥) = 𝐿
𝑥→𝑎



Exercise
What is the limit lim 𝑙𝑛(𝑥)?
𝑥→1



lim 𝑙𝑛(𝑥) = 𝑙𝑛(1) = 0
𝑥→1



Exercise
2
𝑥 −3𝑥+2
What is the limit lim 𝑥−2
?
𝑥→2


0
Substituting 2 gives 0
, which is undefined. However, one can try to simplify the polynomial:
2
𝑥 −3𝑥+2 (𝑥−2)×(𝑥−1)
𝑥−2
= 𝑥−2
=𝑥−1
Now, one can do the limit lim 𝑥 − 1 = 2 − 1 = 1
𝑥→2



We say that 𝐿 is the right-sided (left-sided) limit of 𝑓 as 𝑥 approaches from the right (left),
and it is expressed as lim or lim
+ −
𝑥→𝑎 𝑥→𝑎



Exercise
0 𝑖𝑓 𝑥 < 0
What is the limit lim 𝐻(𝑡) where 𝐻(𝑡) = { 1 𝑖𝑓 𝑥 ≥ 0
𝑡→0



If the 0 is approached from the negative side, the limit returns 0. However, if 0 is approached
from the positive side, the limit equals 1. Thus, we cannot find a defined value 𝐿. In this case,
lim = 1 and lim = 0.
+ −
𝑡→0 𝑡→0



Let 𝑓(𝑥) be a function and suppose it is defined at all points in an open interval containing a
point 𝑥 = 𝑎. then lim 𝑓(𝑥) exists and equals 𝐿 if and only if both one-sided limits both exist
𝑥→𝑎
and equal 𝐿.