Business Statistics: Chapter 6; Discrete Probability Distributions
Probability Distribution
Probability Distribution: defines/describes likelihoods for a range of possible outcomes. Lists all outcomes of an experiment and the probability of each.
Characteristics of a Probability Distribution:
1.) Probability of an outcome is between 0 and 1, inclusive
2.) Outcomes are mutually exclusive. If one occurs, any others cannot occur.
3.) List of outcomes is exhaustive. Sum of all probabilities of outcomes is equal to 1.
Probability Distribution Example: probability of x is P(x).
Number of heads, x Probability of Outcome, P(x)
0 1/8= .125
1 3/8= .375
2 3/8= .375
3 1/8= .125
Total 8/8= 1
Random Variables
Random Variables: when in an experiment of chance, outcomes occur randomly. Measured or observed as the result of an experiment when, by chance, variables may have differing values.
Random Variable Examples: rolling a die, number of employees absent from work, hourly wage of a sample of 50 workers, number of broken lightbulbs produced in an hour, number of drivers charged with DUIs.
Discrete Random Variable: random variable can only assume certain clearly separated values. May be a fraction or decimal so long as values have distance between them.
Continuous Random Variable: random variable may assume an infinite number of values in a given range.
Mean, Variance, Standard Deviation of Discrete Probability Distributions
Mean: value of a random variable to represent the central location of probability distribution. How to find:
Multiply each value of random variable, x, by its probability of occurrence, then add these products.
Variance: describe amount of spread in a distribution. How to find:
Subtract the mean from each value of the random variable, then square each difference
Multiply each squared difference by its probability
Sum resulting products to find the variance
Standard Deviation: how to find:
Square root of the variance
Binomial Probability Distribution
Binomial Probability Distribution: a common discrete probability distribution. Four requirements;
1.) Only 2 possible outcomes on an experimental trial. Two mutually exclusive categories, success or failure.
2.) Random variable is the number of successes for a fixed and known number of trials.
3.) The probability of success is known and the same for each trial.
4.) Each trial is independent of any other trial. The outcome of one trial does not affect the outcome of any other trial.
How to compute Binomial Probability Distribution:
Use =BINOM.DIST in excel
=BINOM.DIST(number of successes, number in the sample, probability of success, cumulative true/false)
Set cumulative to true to find probability up to an including the number of successes. Set to false to find only that specific number of successes.
Binomial Distribution Mean= number in sample x probability of success
Binomial Distribution Variance= mean(1-probability of success)
Poisson Probability Distribution
Poisson Probability Distribution: describes number of times some event occurs in a specified interval. Based on two assumptions…
1.) Probability is proportional to the length of the interval, meaning the longer the interval, the larger the probability.
2.) Intervals are independent, meaning the number of occurrences in one interval doesn’t affect the other intervals
Characteristics of Poisson Probability Distribution:
1.) Random variable is number of times some event occurs in a defined interval
2.) Probability of event is proportional to size of interval
3.) Intervals do not overlap and are independent
How to find Poisson Probability Distribution:
Use =HYPGEOM.DIST(# of successes in sample, # in sample, # possible successes, # in population, cumulative true/false)
Poisson Probability Distribution Mean and Variance: equal to each other. Both found by number in sample x probability of success.