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Stats 309 Probability

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Probability- long run relative frequency. The proportion of times the outcome would occur in a very long series of repetitions
Law of large numbers- the long run frequency of repeated events eventually produces the true relative frequency. This is called the empirical probability or objective probability. When we can’t repeat events the probability of an event is called subjective or personal probability.
If A and B are disjoint, P(A or B)= P(A) + P(B)
General addition rule- P(A or B) = P(A) + P(B) – P(A and B)
If A and B independent- P(A and B) = P(A) * P(B)
Discrete RV- takes on one of a list of possible values
Continuous RV- takes on any value in an interval
Volatility drag = ½ (annual variance)
Long-run return= expected annual return – volatility drag
OR expected annual return – (annual variance)/2
Z-score = X - mean/SD
Bernoulli Requirements
• Two possible outcomes
• Fixed probability of success
• Independence
Binomial requirements- model for categorical variables
• Total number of observations, n, is fixed
• The outcomes of all observations are independent
• Each observation falls into just one of 2 categories
• All observations have the same probability of success
It is okay to ignore dependence if the trials make up less than 10% of the population
P hat is a statistic – p(hat)= X/n
Bernoulli Distribution- simulates one sample and see the individual successes and failures
Binominal Distribution- simulates several samples and see the number of successes and failures in each sample
Sampling error- the difference between sample proportions and the true proportion. This is the sample variability from one sample to the next.
Data AVERAGE STDEV.S
Discrete random variables X1p1+ xnpn Sqrt(var)
Bern Sum P Sqrt(pq)
Binomial, X is # successes in n trials Sum N*p Sqrt(np*q)
Sample proportion p(hat)=X/n Sum P(hat) Sqrt(pq/n)
Sample mean x(bar) Sum X(bar) SD/sqrt(n)
Sampling distribution SD- sqrt(p*q/n)
Normal Model requirements-
• Independence
• Randomization
• 10% condition
• 10 success/failures
Statistic- an observe characteristic of a sample
Parameter- a characteristic of the population
CLT requirements-
• Independence
• Randomization
• 10%
• Large enough sample- 10*(SKEW^2) and 10 * KURT
Exactly- not cumulative
At most- cumulative
At least- cumulative and complement rule
Binominal model- SD = sqrt(n*p*q)
Success/failure only applies to proportions

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