Probability A // Flashcards
Experiment
Any process in which the outcome is uncertain and cannot be predicted in advance
Sample space (Ω)
A set where elements - called sample points - each correspond to a possible outcome of the experiment
Event
A property the outcome of the experiment may or may not have
Random variable
Any quantity we may measure whose observed value depends on the outcome of the experiment
Probability P(A)
(number of sample points in A)/(number of sample points in Ω)
Probability measure
A function A → P(A) for events A ⊆ Ω satisfying:
1. for every event for which P(A) is defined, 0 ≤ P(A) ≤ 1, with P(∅) = 0 and P(Ω) = 1
2. for any finite or countably infinite sequence or mutually disjoint events A1, A2, A3, … with every P(Ai) defined we have
P(∪(i=1,n)Ai) = Σ(i=1,n)P(Ai)
P(∪(i=1,∞)Ai) = Σ(i=1,∞)P(Ai)
(additive property)
Binomial distribution B(n,p)
If X has this distribution, then P(X=k) = (n k)p^k(1-p)^(n-k) for k=0,1,…,n
Poisson distribution Po(λ)
If X has this distribution, then P(X=k) = e^(-λ)λ^k/k!
Normal density function
φ(x) = e^(-x^2/2)/√(2π)
Stirling’s n! formula
n! ~ √(2πn)n^ne^-n
Conditional probability of B given A, P(A) ≠ 0
P(B|A) = P(A ∩ B)/P(A)
Law of total probability
If A1,…,An is a partition of a sample space Ω with P(Ai) > 0 for each I, then for any event B ⊂ Ω we have P(B) = Σ(i=1,n) P(Ai)P(B|Ai)
Bayes formula
If A1,…,An is a partition of Ω with P(Ai)>0 for all i, and B a further event with P(B)>0, then for each j
P(Aj|B) = P(B|Aj)*P(Aj)/(Σ(i=1,n)P(B|Ai)P(Ai))
Independent
Two events A and B are called independent if knowledge A has occurred does not affect the probability of B occurring (and vice versa)
Mutually independent events
Events E1,E2,…,En,… are mutually independent if for all choices of i1,i2,..,in all distinct, then P(Ei1 ∩ Ei2 ∩ … ∩ Ein) = P(Ei1) * P(Ei2) * … P(Ein)
Mutually independent rvs
R.vs X1, X2, …, Xn, … are mutually independent r.vs if whenever B1, B2, …, Bn, … are subsets of ℝ the events {X1 ∈ B1}, {X2 ∈ B2), …, {Xn ∈ Bn} are mutually independent events
Partition of a sample space
(Ai) is a partition if ∪(i=1,n)Ai = Ω, Ai ∩ Aj = ∅ if i ≠ j
Mutually disjoint events A1, A2,…
Ai ∩ Aj = ∅ for i ≠ j
Fundamental multiplication rule for counting elements in a set
A number k choices are to be made. if there are m1 possibilities for the first choice, m2 possibilities for the second choice through to mk for the kth choice, and those may be combined freely, then the total number of possibilities for the procedure is m1m2…*mk
Weak law of large numbers
Fix p ∈ (0,1), let X have the binomial dist with parameters n and p, then for any ε > 0, P(np - nε < X < np + nε) → 1 as n → ∞
Properties of any prob measure
- P(Aᶜ) = 1 - P(A)
- If A ⊂ B then P(A) ≤ P(B)
- For all A,B,
P(A ∪ B) = P(A) + P(B) - P(A ∩ B) - inclusion-exclusion formula for any Ai:
P(∪(i=1, n)Ai) = Σ(i=1, n)P(Ai) -
Σ(i1
Central limit theorem for binomial
Fix p ∈ (0,1), let Xn have the binomial dist with parameters n and p. Then as n → ∞
P(np + z1√(np(1-p)) ≤ Xn ≤ np + z1√(np(1-p)))
→ ∫(z1, z2)φ(x)dx