May 16, 2020
Week 1: Data Science Math Skills
In the first week, I learned three lessons which are:-
- Sets and What They’re Good For
- The infinite World of Real Numbers
- That Jagged S Symbol
So, In the first lesson, the things I learned are:
What is Set?
Cardinality
Intersections
Unions
Medical Testing Example (test and cardinality)
Venn Diagrams
Inclusion-Exclusion Formula
Set
A set is a collection of things, in general set is a collection of elements
for example : A = {1,2,3,4} , the 1, 2, 3 and 4 are the elements of set A
- 1 ∈ A ( means 1 is an element of A)
- 5 ∉ A ( means 5 is not an element of A)
Cardinality
Cardinality means the size of a set means the number of elements in it.
for example: A = {2,3,5,1,2,3} and B = {4,1,2}
- The cardinality of |A| = 6 (because there are 6 elements in A)
- The cardinality of |B| = 3 (because there are 3 elements in B)
Intersections
Intersection simple means the elements which are present in both sets.The symbol for intersection is ∩ which is known as intersects and in general “and”
for example : A = {2,3,4,5,6}, B = {2,4,8,9} and C = {7,8,9}
A ∩ B = {2,4}
B ∩ C = {9}
A ∩ C = ∅
A ∩ C = ∅ (∅, is known as the empty set)Syntax : A ∩ B { x : x ∈ A and x ∈ B}
Unions
Unions simple means the elements which are present in either set.The symbol for union is ∪ which is known as union and in general “or”
for example : A = {2,3,4,5,6}, B = {2,4,8,9}
A ⋃ B = {2,3,4,5,6,8,9}
B ⋃ C = {2,4,8,9,7}
Syntax : A ⋃ B { x : x ∈ A or x ∈ B}
Medical Testing Example
As a part of the course I also learned about a medical testing example. This medical example is best example to understand, how we can implement sets in real world. So, one think to note that VBS means “very bad syndrome” in this example.
let X = set of people in a clinical trial
S = { x ∈ X : x has VBS}, “S for Sick”
H = {x ∈ X : x does not have VBS}, “H for Healthy”
S ∩ H = ∅ (No one person is both S and H)
S ∪ H = X ( either person have VBS or not)Test
Now, lets talked about test
P = {x ∈ X : x tests positive for VBS}, “P for Positive”
N = { x ∈ X: x tests negative for VBS}, “N for Negative”
P ∩ N = ∅ (No one tests is both P and N)
P ∪ N = X (Either test is P or N)- S ∩ P are True Positive
- H ∩ N are True Negative
- S ∩ N are False Negative
- H ∩ P are False Positive
Cardinality (size)
|S|
─── = proportion of people who have VBS
|X||H|
─── = proportion of people who do not have VBS
|X||S∩P|
───── = true positive rate
|X||H∩N|
───── = true negative rate
|X||H∩P|
───── = false positive rate
|X||S∩N|
───── = false negative rate
|X|
Venn Diagrams
Venn Diagrams is the way to visualize sets
for example:
- A = {1,5,10,2}
2. A = {1,9,6,5,7} and B = {3,4,5,7,8}
Inclusion-Exclusion Formula
According to the formula,
|A ⋃ B| = |A| + |B|- |A ∩ B|If we apply this in the upper example Fig 1.2 Multiple Sets
|A ⋃ B| = 8, {1,9,6,3,4,8} |A| = 5, {1,9,6,5,7}|B| = 5, {3,4,8,5,7}|A ∩ B| = 2, {5,7}|A ⋃ B| = |A| + |B|- |A ∩ B|8 = 5 + 5–28 = 10 -28 = 8
In the second lesson, the things I learned are:
Real Numbers : Integers and rational numbers
Absolute value
Intervals and Interval Notation
Real numbers are represented by ℝ, graph of ℝ is mention in fig 2.1
Integers are represented by ℤ for example {..,-3,-2,-1,0,1,2,3,…}
Rational Number are also known as integers they can be written as a/b where b is a non-zero denominator whereas Irrational Number are the numbers which cannot be written as simple fraction ex: pi
Absolute Value
For any x ∈ ℝ,
|x| = x, if x is non-negative|x| = -x, if x is negative
for example :
|1.2| = 1.2
|-1.2| = 1.2 = -(-1.2)
Inequalities
a < b “ a is less than b”a > b “a is greater than b”a ≤ b “a is less than or equal to b”a ≥ b “a is greater than or equal to b”a << b “a is much much lesser than b”
Closed intervals
In closed intervals square brackets [] are used ..
for example:
[2,3] = {x ∊ ℝ : 2 ≤ x ≤ 3}
Open intervals
In open intervals parentheses () are used ..
for example:
(2,3) = {x ∊ ℝ : 2 < x < 3}
Half-open intervals
In Half-open intervals both open and closed intervals are used ..
for example:
1.(2,3] ={x ∊ ℝ : 2 < x ≤ 3}
2.[2,3) = {x ∊ ℝ : 2 ≤ x < 3}
In the third lesson, the things I learned are:
Sigma Notation
Distributive and Commutative property
Summation of constants
Mean and Variance
Sigma Notation
Sigma is represented by Σ, some example are:
for example:
i is just an “dummy indices”, that is used as a counter.Distributive Property
This is due to the distributive property: a(b+c) =ab+ac
Commutative Property
This is due to the commutative property: a+b=b+a
Summation of constants
Summation of constants means if the whole value is constant then sum that value to n times
Mean and Variance
To calculate the value of μ
To calculate mean,
To calculate variance and standard deviation,
