Section 06. The Bernoulli distribution is a special case where n=1!
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Transcript of Section 06. The Bernoulli distribution is a special case where n=1!
Poisson
x is the number of events in a period of time Parameters:
λ – all integers
Lambda can be manipulated for periods of time Ex: If λ represents the number of customers in an
hour, then 2λ is the number in two hours, and λ/2 is the number in half an hour.
Binomial
x is the number of total successes Parameters:
n – number of trials p – probability of success on one trial
denoted
The Bernoulli distribution is a special case where n=1!
Geometric
x is the number of failed trials until first success
Parameters: p – probability of success on single trial
Negative binomial
x is number of failures until r-th success occurs
Parameters: p – probability of success on one trial r – number of successes
The geometric distribution is a special case where r =1!
Multinomial
xi is the number of trials resulting in outcome i
Parameters: n – number of trials p1, p2, …, pk – probability of outcome i on
one trial
Hypergeometric
x is number of Type I objects in selected subset
Parameters: M – number of objects K – number of objects of Type I n – number of objects selected
Likelihood of distributions DEFINITELY know
Uniform Binomial Poisson Geometric
TRY TO know Negative Binomial Hypergeometric
Maybe not Multinomial
Sample Exam #96
A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is .02, independent of all other tourists.
Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 (ticket cost + 50 penalty) to the tourist.
What is the expected revenue of the tour operator?
Sample Exam #30
An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims.
If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?
Sample Exam #67
A baseball team has scheduled its opening game for April 1. If it rains on April 1, the game is postponed and will be played on the next day that it does not rain. The team purchases insurance against rain. The policy will pay 1000 for each day, up to 2 days, that the opening game is postponed.
The insurance company determines that the number of consecutive days of rain beginning on April 1 is a Poisson random variable with mean .6
What is the standard deviation of the amount the insurance company will have to pay?