Business Statistics

Business Statistics 41000 Fact Sheet

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List of topics

1. basic probability 2. mutually exclusive versus independent 3. odds versus probability 4. law of total probability 5. independence 6. Bayes’ rule 7. expected value 8. variance/standard deviation 9. correlation

10. linear combinations 11. normal distribution 12. 68-95-99.7 13. binomial distribution 14. normal approximation to the binomial 15. line of best-fit 16. histograms 17. scatterplots 18. cumulative distribution functions 19. Simpson’s paradox 20. mean reversion Most, put not all, of these topics, are described formally via the formulas below.

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Probability rules

• Probabilities of mutually exclusive events add together. When you roll a die it cannot come up both 2 and 5,

so the probability of it being 2 or 5 is P(2) + P(5).

• The overlap formula: P(A or B) = P(A) + P(B) − P(A and B). • Definition of conditional probability: P(A | B) =

P(A and B) . P(B)

• Compositional form of joint probability: P(A and B) = P(A | B)P(B) • Law of total probability:

P(A) = P(A and B) + P(A and not-B) = P(A | B)P(B) + P(A | not-B)P(not-B).

This generalizes to any number of mutually disjoint events Bj , which we can express with summation notation: P(A) =

j

P(A and Bj ) P(A | Bj )P(Bj ).

j

=

• Putting the compositional form of joint probability together with the definition of conditional probability leads to

Bayes’ rule: P(A | B) = P(B | A) P(A and B) = . P(B) P(B)

• If events A and B are independent: P(A and B) = P(A)P(B) and P(A | B) = P(A).

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Expectations (mean, standard deviation, correlation)

• The expected value of a random variable, denoted E(X) and often shortened as µX , is a weighted sum: µX = E(X) =

x

x P(X = x)

where the sum is taken over all of the values that X can attain.

• More generally, a function of g(X) has expected value

E(g(X)) =

x

g(x)P(X = x).

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For example, if g(X) = X 2 then the formula becomes E(X 2 ) = the formula is E(XY...