Chapter 5 Discrete Probability Distribution. Learning objectives

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1 Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions Distinguish discrete and continuous random variables. 2. Able to compute Epected value and Variance of discrete random variable. 3. Understand: 3.1. Discrete uniform distribution 3.2. Binomial distribution 3.3. Poisson distribution Slide 2 1

2 Random Variables A random variable is a numerical description of the outcome of an eperiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. Slide 3 Eample: JSL Appliances Discrete random variable with a finite number of values Let = number of TVs sold at the store in one day, where can take on 5 values (0, 1, 2, 3, 4 Discrete random variable with an infinite sequence of values Let = number of customers arriving in one day, where can take on the values 0, 1, 2,... We can count the customers arriving, but there is nofinite upper limit on the number that might arrive. Slide 4 2

3 Random Variables Question Random Variable Type Family size Distance from home to store Own dog or cat = Number of dependents reported on ta return = Distance in miles from home to the store site = 1 if own no pet; = 2 if own dog(s only; = 3 if own cat(s only; = 4 if own dog(s and cat(s Discrete Continuous Discrete Slide 5 Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. We can describe a discrete probability distribution with a table, graph, or equation. Slide 6 3

4 Discrete Probability Distributions The probability distribution for discrete random variableis defined by a probability function, denoted by f(, which providesthe probability for each value of the random variable. The required conditions for a discrete probability function are: f( > 0 Σf( = 1 Slide 7 Discrete Probability Distributions Using past data on TV sales, a tabular representation of the probability distribution for TV sales was developed. Number Units Sold of Days f( /200 Slide 8 4

5 Discrete Probability Distributions Graphical Representation of Probability Distribution.50 Probability Values of Random Variable (TV sales Slide 9 Epected Value and Variance The epected value, or mean, of a random variable is a measure of its central location. E( = µ = Σf( The variance summarizes the variability in the values of a random variable. Var( = σ 2 = Σ( - µ 2 f( The standard deviation, σ, is defined as the positive square root of the variance. Slide 10 5

6 Epected Value and Variance Epected Value f( f( E( = 1.20 epected number of TVs sold in a day Slide 11 Epected Value and Variance Variance and Standard Deviation µ ( - µ 2 f( ( - µ 2 f( Variance of daily sales = σ 2 = TVs squared Standard deviation of daily sales = TVs Slide 12 6

7 Random variables # 2 (page 188 # 5 (page 188 In-class Eercise Epected value and variance #16 (page 196 #17 (page 197 Slide 13 Discrete Uniform Probability Distribution The discrete uniform probability distribution is the simplest eample of a discrete probability distribution given by a formula. The discrete uniform probability function is f( = 1/n the values of the random variable are equally likely where: n = the number of values the random variable may assume Slide 14 7

8 Binomial Distribution Four Properties of a Binomial Eperiment 1. The eperiment consists of a sequence of n identical trials. 2. Two outcomes, success and failure, are possible on each trial. 3. The probability of a success, denoted by p, does not change from trial to trial. stationarity 4. The trials are independent. assumption Slide 15 Binomial Distribution Our interest is in the number of successes occurring in the n trials. We let denote the number of successes occurring in the n trials. Slide 16 8

9 Binomial Distribution Binomial Probability Function n ( n f ( = p (1 p n! = p (1 p!( n! ( n where: f( = the probability of successes in n trials n = the number of trials p = the probability of success on any one trial Slide 17 Binomial Distribution Binomial Probability Function n! f ( = p (1 p!( n! ( n n!!( n! Number of eperimental outcomes providing eactly successes in n trials p (1 p ( n Probability of a particular sequence of trial outcomes with successes in n trials Slide 18 9

10 Eample: Evans Electronics 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 net year. Slide 19 Eample: Evans Electronics 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? n! f ( = p!( n! Let: p =.10, n = 3, = 1 n ( p ( 1 3! f (1 = (0.1 (0.9 = 3(.1(.81 =.243 1!(3 1! 1 2 (1 (0.1 (0.9 3(.1( Slide 20 10

11 Tree Diagram 1 st Worker 2 nd Worker 3 rd Worker Prob. Leaves (.1 Stays (.9 Eample: Evans Electronics Leaves (.1 Stays (.9 Leaves (.1 Stays (.9 L ( S (.9 L (.1 L (.1 S (.9 S (.9 L (.1 S ( Slide 21 Eample: Evans Electronics Using Tables of Binomial Probabilities p n Slide 22 11

12 Binomial Distribution Epected Value E( = µ = np Variance Standard Deviation Var( = σ 2 = np(1 p σ = np (1 p Slide 23 Eample: Evans Electronics Epected Value E( = µ = 3(.1 =.3 employees out of 3 Variance Var( = σ 2 = 3(.1(.9 =.27 Standard Deviation σ = 3(.1(.9 =.52 employees Slide 24 12

13 In-class Eercise #26 (page 207 #37 (page 208 Slide 25 Poisson Distribution A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space It is a discrete random variable that may assume an infinite sequence of values ( = 0, 1, 2,.... Slide 26 13

14 Poisson Distribution Eamples of a Poisson distributed random variable: the number of knotholes in 14 linear feet of pine board the number of vehicles arriving at a toll booth in one hour Slide 27 Poisson Distribution Two Properties of a Poisson Eperiment 1. The probability of an occurrence is the same for any two intervals of equal length. 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval. Slide 28 14

15 Poisson Distribution Poisson Probability Function µ e f ( =! µ where: f( = probability of occurrences in an interval µ = mean number of occurrences in an interval e = Slide 29 Eample: Mercy Hospital Patients arrive at the MERCY emergency room of Mercy 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? Slide 30 15

16 Eample: Mercy Hospital MERCY Using the Poisson Probability Function µ = 6/hour = 3/half-hour, = ( f (4 = = ! Slide 31 Eample: Mercy Hospital MERCY Using Poisson Probability Tables m Slide 32 16

17 Eample: Mercy Hospital MERCY Poisson Distribution of Arrivals 0.25 Poisson Probabilities Probability Number of Arrivals in 30 Minutes actually, the sequence continues: 11, 12, Slide 33 Poisson Distribution A property of the Poisson distribution is that the mean and variance are equal. µ = σ 2 Slide 34 17

18 Eample: Mercy Hospital MERCY Variance for Number of Arrivals During 30-Minute Periods µ = σ 2 = 3 Slide 35 In-class Eercise #38 (page 211 #41 (page 212 Slide 36 18

19 End of Chapter 5 Slide 37 19

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