Study Guide on Probability
(e.stat sections are noted in parenthesis)
Questions to ask at the beginning:
- What is the random variable I will model with probability?
- Is the random variable discrete or continuous?
- Are the events statistically independent? Are they mutually exclusive?
- Can I more easily compute the probability of a complement?
- Do events occur at a steady pace (in probability) over time or space?
- Is the pattern of events uniform, skewed, or symmetric-and-bell-shaped?
- Does the setting provide a prior probability to be updated by new information?
When a specific probability distribution function (PDF) applies in a given setting, using it will simplify calculation of probability. When a specific PDF doesn’t apply, we can fall back on the basic axioms and tools for counting. We can also try simulation.
Axioms and Counting (05-03)
The probability of all possible outcomes of a process is one, a certainty.
The probability of the null set is zero.
The probability of any outcome falls between zero and one.
The probability of the complement of an event is one minus the probability of the event.
The Addition Rule (05-12)
For mutually exclusive events
The Multiplication Rule(05-14)
For statistically independent events
Counting (09-06)
To count the number of events that occur together, A and B, multiply.
To count the number of events that are alternatives, A or B, add.
To count the number of selections in which the order of selection matters, use permutations. PERMUT(n,k) (05-17)
To count the number of selections in which the order of selection does not matter, use combinations. COMBIN(n,k) (05-18)
Expectations
To summarize the pattern of events that occur with probability, compute the expected value and standard deviation.
The expected value is the weighted average of outcomes with probability as the weights. The expected value is also called the mean. (06-04)
The standard deviation is the sum of the squared deviations from the mean, weighted by probability. (06-06)
Probability Functions
Chapter 7 of e.stat presents seven PDFs, four of them describe discrete events and three describe continuous events.
Discrete PDFs
Binomial:To find the probability of x successes in n statistically independent trials with π probability of success on each trial. (07-04)
Pascal:To find the probability that n statistically independent trials with probability π of success on each trial are necessary to achieve x successes. (07-06)
Hypergeometric:To find the probability of x successes in n random draws without replacement from a finite population of size N with A successes in the population. (07-08)
Poisson:To find the probability of x successes in an interval of time when a steady flow has average success of per interval. (07-10)
Continuous PDFs
Uniform:To find the probability of events occurring within limits when the probability is proportional to the length of the overall interval, applicable, for example, to inventory problems. (07-12)
Exponential:To find the probability that the time between arrivals when the number of arrivals follows a Poisson process. (07-13)
Normal:To find the probability that an event x falls within an interval when the probability density follows a specific bell-shaped curve with mean, , and standard deviation, . (07-14)
To use the standard normal, transform x to z:
Bayes
To update a prior probability with new information, use Bayes Rule. (09-06)