1. Probability

Question 1

What does statistics involve?

Answer 1

• mathematics
• calculations of numbers
• relies heavily on how the samples are chosen
• relies on how the statistics are interpreted
• the numbers maybe right but the interpretation maybe wrong

Question 2

What are statistics?

Answer 2

• not only facts and figures
• range of techniques and procedures for:
• -analysing
• -interpreting
• -displaying
• -making decisions based on data

Question 3

Probability

Definition

Answer 3

• probabilities are numbers assigned to events, they must satisfy the following properties:
  i) P(Ω) = 1
  ii) P(A)≥0, for all A⊂Ω

Question 4

Ω

Definition

Answer 4

-represents the sample space, the space of all possible outcomes

Question 5

Disjoint Events Probabilities

Answer 5

-if A1 and A2 are disjoint, then:

P(A1∪A2) = P(A1) + P(A2)

Question 6

What follows from disjoint event probabilities?

Answer 6

-for disjoint events:
P(A1∪A2) = P(A1) + P(A2)
-it follows that:
i) P(A^c) = 1 - P(A)
ii) P(∅) = 0
iii) P(A)≤P(B) whenever A⊂B

Question 7

Conditional Probabilities

Answer 7

-the conditional probability of A given that B is known to have occurred is:
P(A|B) = P(A∩B) / P(B), P(B)>0

Question 8

Law of Total Probability

Answer 8

-let B1,B2,…,Bn be a disjoint collection of sets each having positive probability whose union is all of Ω, then:
P(A) = Σ P(A|Bi) P(Bi)

Question 9

Bayes’ Rule

Answer 9

-if in addition to the law of total probability conditions P(A)>0, then:
P(Bj|A) = P(A|Bj)P(Bj) / P(A)

Question 10

Independence

Informal Definition

Answer 10

-events A and B are independent if knowing whether or not A has happened gives no information about whether or not B has happened and vice versa
-i.e. P(A|B) = P(A) and P(B|A)=P(B)
-given that:
P(A∩B) = P(A|B)P(B)
-then we arrive at the formal definition of independence:
P(A∩B) = P(A)P(B)

Question 11

Independence

Formal Definition

Answer 11

• if:
  i) P(A)>0
  ii) P(A∩B) = P(A)P(B)
• then A and B are independent

Question 12

Random Variable

Definition

Answer 12

-the random variable X on a sample space Ω is a real-valued function that assigns to each sample point ω∈Ω a real number X(ω)