By Ian Jordaan
1. Uncertainty and decision-making -- 2. the idea that of chance -- three. Distributions and expectation -- four. the concept that of software -- five. video games and optimization -- 6. Entropy -- 7. Mathematical features -- eight. Exchangeability and inference -- nine. Extremes -- 10. possibility, defense and reliability -- eleven. facts and simulation -- 12. end
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Extra info for Decisions under uncertainty : probabilistic analysis for engineering decisions
6 Terminology, notation and conventions Most of what will be said in the following arises from conventions suggested in many cases by de Finetti. These changes assist considerably in orienting oneself and in posing problems correctly, making all the difference between a clear understanding of what one is doing and a half-formed picture, vaguely understood. As noted previously, by far the most common problem arises from the fact that much previous experience in probability theory results from dealing with ‘repetitive random phenomena’: series of tosses of a coin, records of wind speeds, and in general records of data.
A similar bet could be used to define Pr( E), ˜ for a contract which gives us the quantity ‘s’ if E˜ occurs. 3(a) and (b) that the bet on E˜ (in (b)) is just the negative of the bet on E˜ when Pr(E) was defined (in (a)). 3(b) is equivalent to selling a bet on E for ‘ ps’ with a fine of ‘s’ (rather than a reward) if E occurs. 3(a) and (b) we have accepted that p + p˜ = 1. If this is not the case, what happens? Let us then assume that ˜ are assigned to E and E˜ respectively. 3. 4. The assignment of probability means that the assignor would be willing to pay $80 and $30, a total of $110, for bets on E and E˜ respectively, and receive $100 in return should either happen, a sure loss of $10!
S∈ / S; s is not a member of S. 5(b). 5(c). The word ‘or’ is interpreted as meaning those sample points that are in A and B or both A and B. A ∪ B is sometimes written as A + B, but this contradicts our logical notation so that we shall not use this notation. 5. 5(c). The intersection A ∩ B is sometimes written as AB, which is consistent with our logical notation. 1. 2. 3. 4. For sets and for logical operations ∨ and ∧, both union and intersection follow the associative, commutative and distributive rules, for example A ∪ B = B ∪ A, (A ∩ B) ∩ C = A ∩ (B ∩ C), and so on.