STAT 2120, Fall 2012: Notes on Topic 5 Conditional probability: • A conditional probability, 󰇛|󰇜, gives the probability of some event, , under the condition that some other event, , has definitely occurred. • The general multiplication rule is 󰇛 and 󰇜󰇛󰇜󰇛|󰇜. o This rule extends to multiple events as 󰇛 and  and 󰇜󰇛󰇜󰇛|󰇜󰇛| and 󰇜, etc. o Rearranging this rule yields 󰇛|󰇜󰇛 and 󰇜󰇛󰇜⁄, which serves as a definition of conditional probability. o If  and  are independent, then 󰇛|󰇜󰇛󰇜, which is quickly derived from the previous property 󰇛 and 󰇜󰇛󰇜󰇛󰇜. • A tree diagram is convenient way of organizing one’s thinking when working with conditional probabilities. o Each branch extending from some event to one of possibly many other events represents a segment of a possible path through the stages of a problem. It is labeled by the conditional probability of the latter event given the former. o The conditional probabilities associated with all of the branches extending from the same event must sum to one. o Each complete path from the first to the last stage of a problem represents the overlap of the events along that path. Its probability is calculated by multiplying the corresponding conditional probabilities. o The probability of some event at the final stage of the problem is calculated by adding the product of probabilities along all complete paths that lead to that event. • Bayes’s rule is: 󰇛|󰇜󰇛|󰇜󰇛󰇜/󰇝󰇛|󰇜󰇛󰇜󰇛|󰇜󰇛󰇜󰇞. o It is often easier work with the definition of conditional probability, while manipulating probabilities in a tree diagram, than work with the formula for Bayes’s rule. o The numerator of Bayes’s rule reflects the multiplication of conditional probabilities along a tree diagram’s complete path through  and then . o The denominator of Bayes’s rule reflects the addition of probabilities of all the complete paths in a tree diagram that lead to event . Binomial distributions: • The binomial distributions provide a theoretical model for count data having a fixed maximum. • The binomial setting is defined as follows. o A fixed number, , of trials (i.e., chance happenings) are observed. o The trials are independent. That is, knowing the outcome of any one trial will not affect the probabilities governing any other trial. o Each trial has the same two possible outcomes, whose generic labels are S (for “success”) and F (for “failure”). o The success probability, 󰇛󰇜, is the same for each trial. • Some properties of the binomial setting are: o The sample space consists of 2 possible outcomes, corresponding to the number of possible length- sequences of S and F. o Each possible outcome has probability #󰇛1󰇜#, where # and # count the respective number of S and F in  trials. o There are 󰇡󰇢 possible outcomes with S appearing exactly  times. • The random variable, , that counts the number of S in the binomial setting is called a binomial random variable and is said to have a binomial distribution. • Some properties of the probability model for a binomial random variable, , are: o The sample space consists of 1 possible outcomes, 󰇝0,1,…,󰇞. o Probabilities are assigned as 󰇛󰇜󰇡󰇢󰇛1󰇜. These are sometimes called binomial probabilities. o The mean and standard deviation are  and 󰇛1 󰇜. • Some approaches for finding binomial probabilities are: o Use the formula 󰇛󰇜󰇡󰇢󰇛1󰇜. o Use a binomial table, such as that on p. 325. o Use the Excel function =binomdist(,,,0) for 󰇛󰇜 or =binomdist(,,,1) for 󰇛󰇜. The latter are called cumulative probabilities. o Use a Normal approximation, 󰇛󰇜 󰇛󰇜󰇛1󰇜⁄. A rule of thumb is to apply the Normal approximation when 10 and 󰇛1󰇜10.Poisson distributions: • The Poisson distributions provide a theoretical model for open-ended counts. • The Poisson setting is defined as follows: o “Success points” are counted within a fixed region or time-interval, etc., which is a continuum and may be subdivided into arbitrarily small “units of measure.” o Counts of success points are independent between any nonoverlapping units of measure. o The mean count of success points in any unit of measure is proportional to its size. o The probability of two or more success points in the same unit of measure becomes arbitrarily small as the size of the unit shrinks. • The random variable, , that counts the number of success points in the Poisson setting is called a Poisson random variable and said to have a Poisson distribution. • Some properties of the probability model for a Poisson random variable, , are: o The sample space is 󰇝0,1,…󰇞, which is infinite, but may still be counted; thus,  is a discrete random variable. o Probabilities are assigned as 󰇛󰇜!⁄, where  is the mean count of success points and   2.71828 is the base of the natural logarithms. These are sometimes called Poisson probabilities. o The mean and standard deviation are  and √. • Some approaches for finding Poisson probabilities are: o Use the formula 󰇛󰇜!⁄. o Use the Excel function =poisson(,,0) for 󰇛󰇜 or =poisson(,,1) for cumulative probabilities 󰇛󰇜. • Some points to consider when pondering the Poisson distribution as a model for data: o If  and  are Poisson random variables counting the success points in nonoverlapping regions or time-intervals, etc., then   is a Poisson random variable with mean  . o If  is the mean count of success points per unit of space or time, then  is the mean count of success points