Statistical Applications Binominal and Poisson’s Probability distributions E ( x ) =  =  xf ( x )

Learning Objectives l Evaluate discrete probability distributions from realistic data l Use the binominal distribution to evaluate simple probabilities l Evaluate probabilities using the Poisson’s distribution l Answer examination type question pertaining to these distributions After the session the students should be able to:

Recap- Types of data Discrete (A variable controlled by a fixed set of values) Continuous data (A variable measured on a continuous scale ) These data may be collected (ungrouped) and then grouped together in particular form so that can be easily inspected But how would we collect data?

Simple random sampling Stratified sampling Cluster sampling Quota sampling Systematic sampling Mechanical sampling Convenience sampling Recap: Sampling Techniques

Random Variables A random variable is a numerical description of the A random variable is a numerical description of the outcome of an experiment. outcome of an experiment. A random variable is a numerical description of the A random variable is a numerical description of the outcome of an experiment. outcome of an experiment. A discrete random variable may assume either a A discrete random variable may assume either a finite number of values or an infinite sequence of finite number of values or an infinite sequence of values. values. A discrete random variable may assume either a A discrete random variable may assume either a finite number of values or an infinite sequence of finite number of values or an infinite sequence of values. values. A continuous random variable may assume any A continuous random variable may assume any numerical value in an interval or collection of numerical value in an interval or collection of intervals. intervals. A continuous random variable may assume any A continuous random variable may assume any numerical value in an interval or collection of numerical value in an interval or collection of intervals. intervals.

Examples Question Random Variable x Type Familysize x = Number of dependents reported on tax return reported on tax returnDiscrete Distance from home to store x = Distance in miles from home to the store site home to the store site Continuous Own dog or cat x = 1 if own no pet; = 2 if own dog(s) only; = 2 if own dog(s) only; = 3 if own cat(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) = 4 if own dog(s) and cat(s) Discrete

Discrete Probability Distributions The probability distribution for a random variable The probability distribution for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable. the values of the random variable. The probability distribution for a random variable The probability distribution for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable. the values of the random variable. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or equation. with a table, graph, or equation. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or equation. with a table, graph, or equation.

Discrete Probability Distributions cont… The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), which provides probability function, denoted by f ( x ), which provides the probability for each value of the random variable. the probability for each value of the random variable. The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), which provides probability function, denoted by f ( x ), which provides the probability for each value of the random variable. the probability for each value of the random variable. The required conditions for a discrete probability The required conditions for a discrete probability function are: function are: The required conditions for a discrete probability The required conditions for a discrete probability function are: function are: f ( x ) > 0  f ( x ) = 1

Discrete Uniform Probability Distribution The discrete uniform probability function is The discrete uniform probability function is f ( x ) = 1/ n where: n = the number of values the random variable may assume variable may assume the values of the random variable random variable are equally likely are equally likely

Relative frequency Say a shop uses past knowledge to produce a tabular representation of the probability distribution for TV sales: Number Number Units Sold of Days Units Sold of Days x f ( x ) x f ( x ) /200

Mean Value The or mean or expected value, of a random variable is a measure of its central location. expected number of TVs sold in a day x f ( x ) xf ( x ) x f ( x ) xf ( x ) E ( x ) = 1.20 E ( x ) = 1.20 E ( x ) =  =  xf ( x )

Mean cont… Graphical Representation of Probability Distribution Values of Random Variable x (TV sales) ProbabilityProbability

Variance & Standard deviation The variance summarizes the variability in the values of a random variable. Var( x ) =  2 =  ( x -  ) 2 f ( x ) NOTE: The standard deviation, , is defined as the positive NOTE: The standard deviation, , is defined as the positive square root of the variance. square root of the variance. NOTE: The standard deviation, , is defined as the positive NOTE: The standard deviation, , is defined as the positive square root of the variance. square root of the variance.

Binomial Distribution Four Properties of a Binomial Experiment 3. The probability of a success, denoted by p, does not change from trial to trial. not change from trial to trial. 3. The probability of a success, denoted by p, does not change from trial to trial. not change from trial to trial. 4. The trials are independent. 2. Two outcomes, success and failure, are possible on each trial. on each trial. 2. Two outcomes, success and failure, are possible on each trial. on each trial. 1. The experiment consists of a sequence of n identical trials. identical trials. 1. The experiment consists of a sequence of n identical trials. identical trials. stationaryassumption

Of interest is the number of success occurring in n trials Let x be the number of successes Binomial Probability Function where: where: f ( x ) = the probability of x successes in n trials f ( x ) = the probability of x successes in n trials n = the number of trials n = the number of trials p = the probability of success on any one trial p = the probability of success on any one trial Jacob Bernoulli

Binomial Probability Function cont… Evaluation of probabilities using the distribution function: Probability of a particular sequence of trial outcomes sequence of trial outcomes with x successes in n trials with x successes in n trials Probability of a particular sequence of trial outcomes sequence of trial outcomes with x successes in n trials with x successes in n trials Number of experimental outcomes providing exactly outcomes providing exactly x successes in n trials Number of experimental outcomes providing exactly outcomes providing exactly x successes in n trials

Binomial probability function alternative notation Evaluation of probabilities using the distribution function: No of combinations No of combinations Notice the pattern of numbers Notice the pattern of numbers Probability of a particular sequence of trial outcomes sequence of trial outcomes with x successes in n trials with x successes in n trials Probability of a particular sequence of trial outcomes sequence of trial outcomes with x successes in n trials with x successes in n trials Number of experimental outcomes providing exactly outcomes providing exactly x successes in n trials Number of experimental outcomes providing exactly outcomes providing exactly x successes in n trials

Mean and Variance It is useful to note that for a binominal distribution the following are valid: E ( x ) =  = np Var( x ) =  2 = np (1  p ) Expected Value Variance Standard Deviation

Example #1 : n Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year.

Solution: Using the Binomial Probability Function Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Let : p = 0.1, n = 3, x = 1

Exercise #1 A milling machine is know to produce 9% defective components, if a random sample of 5 components are taken, evaluate the probability of no more than 2 components being defective

Exercise #1: Solution Find the required parameters, namely: p=0.09 n=5 X<3 Here you will need to use a little intelligence, i.e.: Now put the numbers into:

Exercise #1: Solution cont… Using this standard formula gives Notice the pattern: Top and bottom equal the top 4+1=5

Further example A machine produces on average 1 defective parts out of 8. 5 samples are collected from this machine. Find the probability that 2 of them are defective. Solution: Notice here the new nomenclature C 2 5

Poisson’s distribution This distributions is named after the famous French mathematician who formulated it: Siméon Denis Poisson A Poisson distributed random variable is often A Poisson distributed random variable is often useful in estimating the number of occurrences useful in estimating the number of occurrences over a specified interval of time or space over a specified interval of time or space A Poisson distributed random variable is often A Poisson distributed random variable is often useful in estimating the number of occurrences useful in estimating the number of occurrences over a specified interval of time or space over a specified interval of time or space It is a discrete random variable that may assume It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ). an infinite sequence of values (x = 0, 1, 2,... ). It is a discrete random variable that may assume It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ). an infinite sequence of values (x = 0, 1, 2,... ).

Poisson’s random variables They can be time dependent or not! Examples of a Poisson distributed random variable: Examples of a Poisson distributed random variable: the number of knotholes in 14 linear feet of the number of knotholes in 14 linear feet of pine board pine board the number of knotholes in 14 linear feet of the number of knotholes in 14 linear feet of pine board pine board the number of vehicles arriving at a the number of vehicles arriving at a toll booth in one hour toll booth in one hour the number of vehicles arriving at a the number of vehicles arriving at a toll booth in one hour toll booth in one hour

Poisson distribution function Just as with the binominal distribution this allows the calculation of probabilities! where: where: f(x) = probability of x occurrences in an interval  = mean number of occurrences in an interval  = mean number of occurrences in an interval e = e =

Poisson’s cumulative distribution function By definition this is given by: Remembering this pattern helps in the evaluation of the required probabilities since each term in the series are respectively: P(X=0), P(X=1), P(X=2), P(X=3)

Example #2: Patients arrive at the Casualty department of a hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening?

Example #2: Solution o Simply use the Poisson’s distribution function:  = 6/hour = 3/half-hour, x = 4 MERCY

Poisson’s distribution cont… n Poisson Distribution of Arrivals Poisson Probabilities Number of Arrivals in 30 Minutes Probability NB: The Poisson’s distribution has the very special property of the mean and variance being equal!  =  2 Also when n>50, i.e. large and np<5, i.e. small. Then this distribution approximates the Binominal.

Exercise #2: A serviceman is “beeped” each time there is a call for service. The number of beeps per hour is Poisson distributed with a mean of 2 per hour. Find the probability that he gets beeped 3 times in the next 2 hours. Solution: The units of interval need to be uniform. So, the mean beep rate will be 4 per 2 hour intervals. Application of the Poisson’s probability function, renders:

Further Example A garage workshop has an expensive machine tool which is used on average 1.6 times per 8-hour day for a four hour period. How many days in 60 day work period is the tool required no more than twice. Hence the required no. of days

Examination type questions 1.A machine is know to produce 10% defective components, if a random sample of 12 components are taken, evaluate the probability of: a) No components being defective [2] b) more than 3 components being defective [3] 2. If jobs arrive at a machine at random average intervals of 10/hr, estimate the probability of the machine remaining idle for a 1.5 hour period [4]. a) State the standard deviation of this distribution [1].

Further examination type questions 3. Over a long period of time it is known that 5% of the total production are below standard. If 6 are chosen at random, evaluate the probability that at least 2 are defective [5]. 4. A machine is known to produce 2% defective components. In a packet of 100 what is the probability of obtaining over 2 defective components [5].

Solutions: 1. Here a simple application of the Binominal is required thus: 2. Here let X be a Poisson RV, thus:

Solutions: 3. This is the binominal model: 4. We could use the binominal here but it is also a Poisson’s approximation with np=100×0.02, thus:

Alternative solution 4: 4. We could use the binominal here also, with n=100 and p=0.02: Note: It’s worth noticing the Poisson’s approximation if it turns up, “less calculations”!

Summary l Evaluate discrete probability distributions from realistic data l Use the binominal distribution to evaluate simple probabilities l Evaluate probabilities using the Poisson’s distribution l Answer examination type question pertaining to these distributions Have we met out learning objectives? Specifically are you able to: