STATISTICS FOR ECO:
SUMMARY
@ECOsummaries
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1
,Statistics summary
6.1 Random experiments
Random experiment: an experiment for which the outcome is determined by chance
Sample space (Ω): set of all N possible outcomes
Venn diagram:
Events: subsets of the sample space
Single events: events with one specific outcome (BBB or AAA or BCB or CBB etc.)
Multiple events: more than one outcome (BBB, AAA and CBB or BCC, ABB and AAC etc.)
Empty event: does not occur (empty set)
6.2 Rules for sets / events
1. A c B A is subset of B if A occurs, then B occurs
2. A-complement complement of A A does not occur
3. A u B union of A and B A or B occurs (or both)
4. A ∩ B intersection of A and B A and B occur
5. A ∩ B = disjoint A and B are disjoint no common outcomes
6.3 Historical definitions of probability
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,* Classical definition of probability:
Requirement: all outcomes of the experiment are equally likely
Properties:
- 0 ≤ P(A) ≤ 1
- P(disjoint) = 0 and P(Ω) = 1
- P(A u B) = P(A) + P(B) If A and B are disjoint!
* Empirical definition of probability:
Requirement: the experiment is independently and identically repeatable
Independently: the experiment don’t influence each other
Identically: the probabilities remain the same
Properties:
- 0 ≤ P(A) ≤ 1
- P(disjoint) = 0 and P(Ω) = 1
- P(A u B) = P(A) + P(B) If A and B are disjoint!
* Subjective definition of probability:
Can be used for all random experiments
Disadvantage: it is subjective
Properties:
- 0 ≤ P(A) ≤ 1
- P(disjoint) = 0 and P(Ω) = 1
- P(A u B) = P(A) + P(B) If A and B are disjoint!
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, 7.1 Basic properties of model P (axioms)
1. P(A-complement) = 1 – P(A)
2. P(disjoint) = 0
3. A c B = P(A) ≤ P(B)
4. P(A) = P(A ∩ B) + P(A ∩ B-complement)
5. /
6. P(A u B) = P(A) + P(B) – P(A ∩ B)
7.2 Rules for counting
1. # orderings of k objects
k! = k factorial e.g. 🡪 4! = 4 * 3 * 2 * 1 = 24
0! = 1 (by definition)
2. # possibilities to choose k objects from m objects
Ordered: in a specific order
With replacement: you can pick the same k for several places
Without replacement: you can only put the same k in 1 place
2a. Ordered, with replacement
2b. Ordered, without replacement
2c. Unordered, without replacement
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SUMMARY
@ECOsummaries
→ 20% discount
1
,Statistics summary
6.1 Random experiments
Random experiment: an experiment for which the outcome is determined by chance
Sample space (Ω): set of all N possible outcomes
Venn diagram:
Events: subsets of the sample space
Single events: events with one specific outcome (BBB or AAA or BCB or CBB etc.)
Multiple events: more than one outcome (BBB, AAA and CBB or BCC, ABB and AAC etc.)
Empty event: does not occur (empty set)
6.2 Rules for sets / events
1. A c B A is subset of B if A occurs, then B occurs
2. A-complement complement of A A does not occur
3. A u B union of A and B A or B occurs (or both)
4. A ∩ B intersection of A and B A and B occur
5. A ∩ B = disjoint A and B are disjoint no common outcomes
6.3 Historical definitions of probability
2
,* Classical definition of probability:
Requirement: all outcomes of the experiment are equally likely
Properties:
- 0 ≤ P(A) ≤ 1
- P(disjoint) = 0 and P(Ω) = 1
- P(A u B) = P(A) + P(B) If A and B are disjoint!
* Empirical definition of probability:
Requirement: the experiment is independently and identically repeatable
Independently: the experiment don’t influence each other
Identically: the probabilities remain the same
Properties:
- 0 ≤ P(A) ≤ 1
- P(disjoint) = 0 and P(Ω) = 1
- P(A u B) = P(A) + P(B) If A and B are disjoint!
* Subjective definition of probability:
Can be used for all random experiments
Disadvantage: it is subjective
Properties:
- 0 ≤ P(A) ≤ 1
- P(disjoint) = 0 and P(Ω) = 1
- P(A u B) = P(A) + P(B) If A and B are disjoint!
3
, 7.1 Basic properties of model P (axioms)
1. P(A-complement) = 1 – P(A)
2. P(disjoint) = 0
3. A c B = P(A) ≤ P(B)
4. P(A) = P(A ∩ B) + P(A ∩ B-complement)
5. /
6. P(A u B) = P(A) + P(B) – P(A ∩ B)
7.2 Rules for counting
1. # orderings of k objects
k! = k factorial e.g. 🡪 4! = 4 * 3 * 2 * 1 = 24
0! = 1 (by definition)
2. # possibilities to choose k objects from m objects
Ordered: in a specific order
With replacement: you can pick the same k for several places
Without replacement: you can only put the same k in 1 place
2a. Ordered, with replacement
2b. Ordered, without replacement
2c. Unordered, without replacement
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