Chapter 5 Discrete Probability Distributions

Probability Experiment A probability experiment is any activity that produces uncertain or “random” outcomes

Random Variable A random variable is a rule or function that translates the outcomes of a probability experiment into numbers.

Table 5.1 Illustrations of Random Variables EXPERIMENTPossible Random VariablesType of Variable Commuting to workTime it takes to get to work Number of red lights on the way Amount of gas consumed Continuous Discrete Continuous Advertising a productNumber of customer responses Number of units sold Discrete Taking inventoryNumber of damaged items found Remaining shelf life of an item Discrete Continuous Playing a round of golfDriving distance off the first tee Number of pars Number of lost balls Continuous Discrete Manufacturing a productAmount of waste produced (lbs.) Number of units finished in an hour Continuous Discrete Interviewing for a jobNumber of rejections Duration of the interview Elapsed time before being hired Discrete Continuous Buying stocksNumber of your stocks that increase in value Amount of sleep lost from worry Discrete Continuous

Discrete Random Variable A discrete random variable has separate and distinct values, with no values possible in between.

Continuous Random Variable A continuous random variable can take on any value over a given range or interval.

Probability Distribution A probability distribution identifies the probabilities that are assigned to all possible values of a random variable.

Producing a Discrete Probability Distribution Step 1: Defining the Random Variable Step 2: Identifying Values for the Random Variable Step 3: Assigning Probabilities to Values of the Random Variable

Figure 5.1 Probability Tree for the Management Training Example J (.7) Jones Fails S (.9) J (.7) S' (.1) Smith Passes Smith Fails Jones Passes S∩J S∩J’ S'∩J S'∩J’ Jones Fails J' (.3) Jones Passes J' (.3) (1) (2) (3) (4) Outcome x P(x)

Probability Distribution for the Training Course Illustration Number of Managers Passing x Probability P(x)

Figure 5.2 Graphing the Management Training Distribution P(x) Number of Managers Passing x

Expected Value for a (5.1) Discrete Probability Distribution E( x ) =  x  P( x )

Distribution Variance (5.2)  2 =

The Binomial Distribution

Figure 5.3 Probability Tree for the Coin Toss Example H (.4) T (.6) H (.4).096 T (.6) H (.4).096 H (.4) T (.6) H (.4) T (.6) Heads Heads 1 Head 2 Heads 1 Head 0 Heads

The Binomial Conditions (1)The experiment involves a number of “trials”— that is, repetitions of the same act. We’ll use n to designate the number of trials. (2) Only two outcomes are possible on each of the trials. This is the “bi” part of “binomial.” We’ll typically label one of the outcomes a success, the other a failure. (3) The trials are statistically independent. Whatever happens on one trial won’t influence what happens on the next. (4) The probability of success on any one trial remains constant throughout the experiment. For example, if the coin in a coin-toss experiment has a 40% chance of turning up heads on the first toss, then that 40% probability must hold for every subsequent toss. The coin can’t change character during the experiment. We’ll normally use p to represent this probability of success.

The Binomial Probability Function (5.4) P (x) =

Expected Value for a (5.5) Binomial Distribution E(x) = nּ p

Variance for a Binomial Distribution (5.6)  2 = n ּpּ(1-p)

Standard Deviation for a (5.7) Binomial Distribution  =

Symmetric Figure 5.4 Some Possible Shapes for a Binomial Distribution Positively Skewed x P(x) x P(x) Negatively Skewed x P(x)

The Poisson Distribution

The Poisson Conditions (1)We need to be assessing probability for the number of occurrences of some event per unit time, space, or distance. (2) The average number of occurrences per unit of time, space, or distance is constant and proportionate to the size of the unit of time, space or distance involved. (3) Individual occurrences of the event are random and statistically independent.

Poisson Probability Function (5.8) P(x) =

Figure 5.5 Graphing the Poisson Distribution for  = 1 mean  =1 P(x) x

Figure 5.6 Matching Binomial and Poisson Distributions x P(x) BINOMIAL n  = 20, p =.10 POISSON  = 2 P(x) x