Maths Techniques 2
Functions
Bijectivity - Both surjective and injective
Injective - Every a maps to a unique b, so no two values are mapped to the same b
Functions must be defined and single valued.
Surjectivity - Every b in B has at least an a mapped to it.
Single Valued - For every a, there must be only one b. But Every b can have multiple a's
Definedness - For every a in A, there must be a B in b so that (a,b) in R
Value table - Lists the input and output of a function for a group of values
R ⊆ A × B or (f: A -> B). A is the domain. B the co-domain. Set of all outcomes is the range
Forward Image - Subset X of A. f(X) ⊆ range(f) ⊆ B
Pre-Image - Subset Y of B. Which values in A get mapped into Y. f-1(Y) ⊆ A. Y !⊆ range(f)
Composing Functions - A -> B and B ->C. Therefore A->C
Probability
Sample Space - The set of all outcomes. Eg Rolling 2 dice -> 36 outcomes
Event - Subset of samples space. Elementary - If its only one value
Positivity - Prob of an event > 0
Totality - All events sum to 1
Disjoint Union - If E and F are disjoint (don't share values) then p(E∪F) = p(E) + p(F)
Complement - p(E) + p(¬E) = 1
Union - p(E∪F) = p(E) + p(F)− p(E∩F)
Probability Space - Non-empty sample space with a probability function
Monotonicity - If B ⊆ A then p(A) = p(B) + p(A\B)
Equiprobable Space - Each outcome has equal probability
Binomial Space - Only 2 outcomes
Independent Experiments - Probability of 2 or more independent events. Eg. Rolling 2 dice. p(AxB) = p(A) x p(B)
Total Correlation - Outcome of event is based entirely on event 1. So theres a f: A -> B
Partly Dependant - Eg. 2 white balls, 1 black. Taking one and not returning it. Probability is based on what the outcome was, but they get updated
Conditional Probabilities - p(B|A) = p(A∩B)/p(A)
Independent Probabilities - p(B|A) = p(B)
Bayes - p(A|B) = p(A) x p(B|A)/p(B) or = p(A)*p(B|A)/p(A)*p(B|A) + p(¬A)*p(B|¬A)
Discrete Random Variables
Random Variable - Combination of a sample space, prob function and number assignment. Basically maps a value to a value in the Real number set.
Discrete - If the range of the random variable is countable
Expected Value - Weighted average value. Notation is E[X]
Standard Deviation - How far away the values tend from the expected value. SD[X] = root(Var[X])
Variance - Var[X] = ∑(r −E[X])^2 * p(X = r)
Poisson - Copy the formula.
Continuous Random Variables
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