5-1 Probability Distributions

• A random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure

• A probability distribution is a description that gives the probability for each value of the random variable. Often expressed in the format a table, formula, or graph.

• A discrete random variable has a collection of values that is finite/countable.

• A continuous random variable has infinitely many values, and the collection of values is not countable.

• Probability histograms are one of the ways to graph a probability distribution

• The expected value of a discrete random variable x is denoted by E, and it is the mean value of the outcomes, so E = mu and E can also be found by evaluating the sum of x * P(x)

• The expected value of a random variable x is equal to the mean mu

• If, under a given assumption, the probability of a particular outcome is very small and the outcome occurs significantly less than or significantly greater than what we expect with that assumption, we conclude that the assumption is probably not correct

5-2 Binomial Probability Distributions

• A binomial probability distributionresults from a procedure that meets these 4 requirements:

  • Procedure has a fixed number of trials

  • The trials must be independent, meaning that the outcome of any individual trial doesn't affect the probabilities in the other trials

  • Each trial must have all outcomes classified into exactly 2 categories, commonly referred to as success and failure

  • The probability of a success remains the same in all trials

• When using a binomial probability distribution, always be sure that x and p are consistent in the sense that they both refer to the same category being called a success

• When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent even though they are actually dependent.

• Significantly low values <= (mu - 2sigma)

• Significantly high values >= (mu + 2sigma)

• Values not significant: between (mu - 2sigma) and (mu + 2sigma)

• For binomial distributions:

  • Mean = mu = np

  • Variance = sigma squared = npq

  • Standard deviation = sigma = root (npq)

5-3 Poisson Probability Distributions

• A Poisson probability distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit.

• Parameters of the Poisson probability distribution:

  • The mean is mu.

  • The standard deviation is sigma = root (mu)

• Properties of the Poisson probability distribution:

  • A particular Poisson distribution is determined only by the mean mu.

  • A Poisson distribution has possible x values of 0, 1, 2... with no upper limit

• Requirements for the Poisson probability distribution:

  • random variable x is the # of occurrences of an event in some interval

  • the occurrences must be random

  • the occurrences must be independent of each other

  • the occurrences must be uniformly distributed over the interval being used

• Requirements for using Poisson as an approximation to binomial:

  • 1. n >= 100

  • 2. np <= 10

• Mean for Poisson as an approximation to binomial: mu = np