Q1: Quantitative Methods Flashcards
Single events
one specific
outcome
Multiple events
more than
one outcome
Union
arba. At least one of A and B occurs
(any element in A or B
Intersection
ir. Both A and B occur
Subset
If A occurs, then B occurs
Disjoint
A and B cannot occur jointly
Partition
The sets D1, D2, …, Ds form a partition if they are mutually disjoint and their union is Omega (entire sample space)
Classical definition of probability
- P(O/) = 0 and P(O/) = 1
- 0 ≤ P(A) ≤ 1 for all events A: 0/𝑁≤𝑁𝐴/𝑁≤𝑁/𝑁
- P(A union B) = P(A) + P(B) if A, B disjoint
all outcomes of an experiment are equally likely
arbitrarily
= randomly
Empirical Definition of Probability
Requirement: the random experiment is independently and identically
repeatable.
In n trials, n(A) is the number of trials where A occurs. The law of large numbers states that:
1) The ratio n/n(A) approaches a constant as 𝒏 → ∞
2) This constant is P(A), same properties hold.
probability of an event is determined by its relative frequency over a large number of trials
Subjective Definition of Probability
Probability P(A) reflects how strongly an individual believes in the future occurrence of event A.
* Can be used for all random experiments
* Disadvantage: it is subjective – different people may obtain different
probabilities for the same event
* Same properties should hold
General Definition of Kolmogorov: basic axioms of a probability model
A probability measure or model P assigns
real numbers P(A) to all events A (subsets of Ω), in such a way that:
(1) P(A) >= 0
(2) P() = 1
(3) If A, B disjoint, then P(AB) = P(A) + P(B)
All three definitions discussed above provide a probability model.
Random drawing
one population element is arbitrarily chosen
Random sampling
randomly drawing n elements from the population
In how many ways can we order k objects?
Number of orderings of k objects.
k!
In how many ways can k objects be chosen from a set of m objects?
Number of possibilities to choose k objects from m objects.
k-tuple: chosen sequence of k objects.
a) ordered, with replacement (=after each draw the chosen object is put back
into the population)
b) ordered, without replacement
c) unordered, without replacement
d) unordered, with replacement (not considered here)
(2a) ordered, with replacement (k objects from the set of m objects)
- m possibilities to choose for the first position
- the first drawn object is replaced back to set m -> again, m possibilities
to choose for the second position, etc - number of possibilities for all k choices equal to m
Total number of orderings: 𝑚^k
(2b) ordered, without replacement (k objects from the set of m objects)
- m possibilities to choose for the first position
- m-1 possibility to choose for the second, m-2 for the third, etc
- m-k+1 possibilities at the last, kth choice
𝑚!/(𝑚−𝑘)!
(2c) unordered, without replacement (k objects from the set of m objects)
- Compared to 2b), groups of k! different orderings become equal
- Need to take the number of orderings from 2b and divide it by k! to get the
unordered number pf k-tuples without replacement
Total number of orderings: 𝑚!
𝑘!×(𝑚−𝑘)!= (𝑚
𝑘) (taip ir atrodo) - Pronounced “m choose k”
- Called binomial coefficients
Conditional probability
Conditional probability of event A given that event B occurs:
𝑃𝐴|𝐵 =𝑃𝐴∩𝐵 / 𝑃(𝐵)
Pronunciation: probability of A given B
Another notation: PB(A)
In this context : P(A) is the prior probability
(before the info B became available)
P(A|B) is the posterior probability
(after the info B became available)
Conditional probability: Rules
Product rule (Notion WEEK 2 (1 ft))
A and B are independent if
if 𝑃𝐴𝐵 =𝑃(A)
~ if the pre-information about the occurrence of B doesn’t influence the probability of A
Bayes’s Rule
Notion WEEK 2
Sample
Some subset of a population
Population
The entirety
Random variable (RV)
function that takes outcome of experiment and returns a measure/value
