1. Probability Flashcards

1
Q

What does statistics involve?

A
  • 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
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2
Q

What are statistics?

A
  • not only facts and figures
  • range of techniques and procedures for:
  • -analysing
  • -interpreting
  • -displaying
  • -making decisions based on data
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3
Q

Probability

Definition

A
  • probabilities are numbers assigned to events, they must satisfy the following properties:
    i) P(Ω) = 1
    ii) P(A)≥0, for all A⊂Ω
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4
Q

Ω

Definition

A

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

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5
Q

Disjoint Events Probabilities

A

-if A1 and A2 are disjoint, then:

P(A1∪A2) = P(A1) + P(A2)

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6
Q

What follows from disjoint event probabilities?

A
-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
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7
Q

Conditional Probabilities

A

-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

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8
Q

Law of Total Probability

A

-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)

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9
Q

Bayes’ Rule

A

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

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10
Q

Independence

Informal Definition

A

-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)

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11
Q

Independence

Formal Definition

A
  • if:
    i) P(A)>0
    ii) P(A∩B) = P(A)P(B)
  • then A and B are independent
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12
Q

Random Variable

Definition

A

-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)

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