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Why do you have to be careful with the geometric and the negative binomial distributions?
There are 2 ways to define each distribution. One way is the number of trails before success (outcome is in terms of trials). The other way is the number of failures before success (outcome is in terms of failures).
What is the PDF for the geometric distribution? (in terms of trials)
What is the PDF for the geometric distribution ? (in terms of failures)
What are the 2 CDFs for the geometric distribution?
- CDF in terms of trials: 1-(1-p)n
- CDF in terms of failures: 1-(1-p)n-1
What are the 2 mean formulas and the variance formula for the geometric distribution?
- Mean in terms of trials: 1/p
- Mean in terms of failures: (1/p)-1
- Variance for both: (1/p2)-(1/p)
True or false: Like the exponential distribution, the geometric distribution has the memoryless property.
True. Knowing that something has occurred (i.e. that a test taker has already answered 4 problems wrong in a row) does not alter the future probability of selecting a right or wrong answer.
If p=.2, what is the probability of getting 3 questions wrong on a multiple choice test before getting the fourth right?
- If you use the number of failures PDF:
- P(x)= p(1-p)3P(x)= .2(.8)3= 10.24%
What is the PDF for the binomial distribution?
Shortcut formulas for the Binomial Distribution.
- Var(x)=np(1-p) or npq where q=1-p
- Mx(t)= (q+pet)n
What is the PDF for the Poisson distribution?
Customers walk into a store at an average rate of 20 per hour. Find the probability that no customers have arrived at the store in 10 minutes.
- Because there are 6 ten-minute intervals in an hour the average number of arrivals= 20/6 or 10/3.
- f(x)= (10/3)0 / 0! * e-10/3
A beach resort buys a policy to insure against loss of revenues due to major storms in the summer. The policy pays a total of $50,000 if there is only one major storm during the summer, a total of $100,000 if there are two major storms, and a total payment of$200,000 if there are more than two major storms.The number of major storms in one summer is modeled by a Poisson distribution with mean of 0.5 per summer.Find the expected premium for this policy during one summer.
- Let N =# of major storms in one summer. N has Poisson distribution with the mean λ=0.5.
- f(n)= and E(x)= x * f(x)
- so the sum of the payouts * the relative probability. 0*e-.5 + 50,000*.5e-.5 + 100,000*.52/2 *e-.5 + 200,000*(1-p: because of the law of total prob) [1-p= 1-(payout 1 + payout 2 + payout 3)=.0144
What is the PDF of the uniform distribution?
What are the key formulas for the uniform distribution? f(x), E(x),
(or var x)?
What are the key formulas for the exponential distribution?
Critical point 1 - Negative Binomial
Negative binomial distribution is like a “negative” version of our familiar binomial distribution. In a binomial distribution, the number of _____ n is fixed and we want to find the probability of having k number of successes in these n trials (so K is the random variable). In contrast, in a negative binomial distribution, the number of ________, k , is fixed and we want to find out the probability that these k successes are produced in n trials (so N is the random variable).
Critical point 2 - Negative binomial
The probability mass function of a negative binomial distribution with parameter (n,k,p) is a fraction of the probability mass function of the binomial distribution with identical parameters (n,k,p) , where n = # of trials, k = # of successes,and p =the probability of success. The fraction is equal to k/n. What does this look like in function form?
A fair die is repeatedly thrown n times until the second 5 appears. What is the probability that n =20? What is the probability of having to throw the die more than 20 times (i.e.n >= 21)? (What does the formula look like?)
A fair die is repeatedly thrown n times until the second 5 appears. What is the probability of having to throw the die more than 20 times (i.e.n >= 21)? What is the easy way to calculate the probability?
- P(throwing the die at least 20 times to get the second 5)=P(having no 5's in 20 trials) + P(having one 5 in 20 trials)
- Notice that this is a binomial distribution:
- So P= C(20,0)(1/6)0(5/6)20 + C(20,1)(1/6)1(5/6)19
If a geometric or negative binomial problem has X=1,2,3,...: then, is it the trial or failure version of this problem?
Trial version. You always have to have at least 1 trial.
If a geometric or negative binomial problem has X=0,1,2,3,...: then, is it the trial or failure version of this problem?
Failure version. Failures can always equal zero (i.e. you get it right the first time)
Critical point 3 - Negative binomial
A negative binomial distribution with parameters (n,k,p) is the sum of k independent identically distributed ________ distribution with parameter p. If you understand this, memorizing the formulas for the mean and variance of a negative binomial distribution becomes easy. You just multiple the formulas by _____.
- So Mean = k * 1/p or k/p
- Var= K(1/p2 - 1/p)
What is the PDF for the normal distribution?
When do you use the negative binomial distribution?
The negative binomial distribution is used when you are finding the number of successes in a sequence of Bernoulli trials before a specified (non-random) number of failures (denoted r) occur.
When do you use the Poisson distribution?
- Poisson distribution is often used to model the occurrence of a random event that happensunevenly (in some time periods it happens more often than other time periods). The occurrences of many natural events can be approximately modeled as a Poisson distribution such as:
- • The number of claims that happen in a given time interval (a month, a year,etc.)
- • The number of hits at a website in a given time interval
- • The number of customers who arrive at a store in a given time interval
- • The number of phone calls (or e-mails) you get in a day
- • The number of shark attacks in one summer
When do you use the exponential distribution?
- Time until the next claim arrives in the claims office.
- Time until you have your next car accident.
- Time until the next customer arrives at a supermarket.
- Time until the next phone call arrives at the switchboard.
The exponential distribution describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. It is the continuous analogue of the geometric distribution.
When do you use the uniform distribution?
The uniform distribution is used when all intervals of the same length on the distribution's support are equally probable.
Imagine a shipment has seven defective parts and ten good parts. You want to find out if you randomly take out five parts, how likely is it to get three defective parts and two good parts. What kind of distribution is this? What is the formula?
- Hypergeometric: remember for the binomial to work, you have to have independent draws.
- The formula is:
- [C(7,3)*C(10,2)] / C(17,5)
Among 200 people working for a large actuary department of an insurance company, 120 are actuaries and 80 are support staff. If 8 people are randomly chosen to attend a brainstorm meeting with company executives, what is the probability that exactly 5 actuaries are chosen to attend the meeting? (Formula)
C(120,5) * C(80,3) / C(200,8)
What does the gamma distribution model?
While exponential distribution models the time elapsed until one random event occurs, gamma distribution models the time elapsed until n random events occur.
Having n random events is the same as having a series of one random event. If a machine malfunctions three times during the next hour, the machine is really having three separate malfunctions within the next hour. As such, gamma distribution is really the sum of n identically distributed exponential random variables.
What is the formula for the CDF for the gamma distribution?
- If you want to the P(X>x)=P( it takes longer than time x to have n random events) just eliminate the 1- from the front of the formula [law of total prob]
What is the formula for the gamma pdf?
The linear combination of 2 or more independent normal variables is distributed?
What is the formula for Z, the standard normal distribution?
What is the Chi-squared distribution?
The Chi-squared distribution is the sum of squared independent standard normal distributions.
What are the mean and variance formulas for the chi-squared distribution?
- Mean=n (equal to the degrees of freedom)
- Variance=2n (variance equal to twice the degree of freedom)
What is the formula for the cov(x,y) and how do you calculate it?