All Formulas Flashcards
How to find P(A) - 4
- Define sample space Ω
- Determine probabilities for each ω in Ω, if all outcomes are equally likely then P(ω) = 1/N
- Determine which outcomes belong to A
- Compute P(A) by Summing all Probabilities of outcomes which belong to A
Addition rule - 1
- P(A u B) = P(A) + P(B) - P(A n B)
Addition rule disjoint events - 3
- Two events are disjoint if they cannot happen at the same time.
- A n B = ø
- P(A u B) = P(A) + P(B)
Complement (and complement rule) - 2
- The complement of A (A bar or Ac) is the set of outcomes which is not in A
- P(A bar) = 1 - P(A)
Conditional probability - 3
- The probability of B occurs after A has occurred
- P(B|A) = P(A n B) / P(A)
- !!! P(B|A) ≠ P(A|B)
Multiplication rule - 1
- P(A n B) = P(A) * P(B|A)
Independence - 2
- Two events A and B are independent if P(A n B) = P(A) * P(B)
- Independence ≠ Disjointness
Complement of at least one - 1
- P(at least one occurence of … ) = 1 - P(no occurence of … )
Simple law of total probability - 2
- A and B are two events
- P(B) = P ( B ∩ A ) + P ( B ∩ A bar ) = P ( B | A ) · P ( A ) + P ( B | A bar ) · P ( A bar )
Bayes’ Theorem - 3
- P(A|B)= P(B|A)·P(A) / P(B|A)·P(A)+P(B|A bar)·P(A bar).
- P(B|A) + P(B bar | A) = 1
- P(B|A) + P(B|A bar ) ≠ 1
Law of total probability - 2
- A1, … , Am is a partition
- P(B) = Sum(m, i = 1) P(B ∩ Ai ) = Sum(m, i = 1) P(B|Ai ) · P(Ai ).
Find probability distribution of discrete random variable - 5
- Determine sample space of probability experiment and probabilities of each outcome
- List numerical values X(ω) for ω ∈ Ω
- For each numerical value x of X find collection of simple events which have particular numerical value x {X = x} = {ω : X(ω) = x}
- use P({w}) determine probability of event {X = x}
P(X =x)= P({ω : X(ω) =x})= sum(ω:X (ω)=x) P({ω} - Tabulate results with left column with all values x of X and a column with probabilities P(X = x)
Expected value (expectation/mean) - 3
- X is a discrete random variable
- Expected value of X is weighted average of the possible values of X
- μ = sum(k, i = 1) xi * P(X = xi)
Variance and SD of X - 2
- σ^2 =Var(X) = sum(k, i = 1) (xi −μ)^2 2 * P (X =xi).
- σ = SD(X) = √Var(X)
Normal distribution formula - 1
- p(x) = 1 / σ(√2π) * e^(-(1/2) * ( (x-μ) /σ )^2
Relation between general normal distribution and standard normal - 2
- If X is normally distributed (mean μ and standard deviation σ) then a shifted and rescaled version Z has a standard normal distribution
- Z = (X - μ) / σ
z score value of x - 2
- Number of standard deviations away from the mean
- z = (x - μ) / σ
Central limit theorem - 3 + 2
- Sample of size n > 30 from population with mean μ and standard deviation σ
- For normal population the sample can be of any size, instead of n > 30
- X Bar, or sample mean has approximately a normal distribution
- mean = μ
- Standard deviation = σ / √n
