1 1 Slide © 2006 Thomson/South-Western Chapter 5 Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution

2 2 Slide © 2006 Thomson/South-Western 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. Random Variables 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.

3 3 Slide © 2006 Thomson/South-Western Let x = number of TVs sold at the store in one day, Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) Let x = number of TVs sold at the store in one day, Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) Example: JSL Appliances n Discrete random variable with a finite number of values

4 4 Slide © 2006 Thomson/South-Western Let x = number of customers arriving in one day, Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... where x can take on the values 0, 1, 2,... Let x = number of customers arriving in one day, Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... where x can take on the values 0, 1, 2,... Example: JSL Appliances n Discrete random variable with an infinite sequence of values We can count the customers arriving, but there is no We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

5 5 Slide © 2006 Thomson/South-Western Random Variables 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

6 6 Slide © 2006 Thomson/South-Western 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

7 7 Slide © 2006 Thomson/South-Western 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: Discrete Probability Distributions f ( x ) > 0  f ( x ) = 1

8 8 Slide © 2006 Thomson/South-Western n a tabular representation of the probability distribution for TV sales was developed. distribution for TV sales was developed. n Using past data on TV sales, … Number Number Units Sold of Days Units Sold of Days x f ( x ) x f ( x ) /200 Discrete Probability Distributions

9 9 Slide © 2006 Thomson/South-Western Values of Random Variable x (TV sales) ProbabilityProbability Discrete Probability Distributions n Graphical Representation of Probability Distribution

10 Slide © 2006 Thomson/South-Western Expected Value and Variance The expected value, or mean, of a random variable The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. The expected value, or mean, of a random variable The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. The variance summarizes the variability in the The variance summarizes the variability in the values of a random variable. values of a random variable. The variance summarizes the variability in the The variance summarizes the variability in the values of a random variable. values of a random variable. The standard deviation, , is defined as the positive The standard deviation, , is defined as the positive square root of the variance. square root of the variance. The standard deviation, , is defined as the positive The standard deviation, , is defined as the positive square root of the variance. square root of the variance. Var( x ) =  2 =  ( x -  ) 2 f ( x ) E ( x ) =  =  xf ( x )

11 Slide © 2006 Thomson/South-Western n Expected Value 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 Expected Value and Variance

12 Slide © 2006 Thomson/South-Western n Variance and Standard Deviation x -  ( x -  ) 2 f(x)f(x)f(x)f(x) ( x -  ) 2 f ( x ) Variance of daily sales =  2 = x TVssquared Standard deviation of daily sales = TVs Expected Value and Variance

13 Slide © 2006 Thomson/South-Western Binomial Distribution n 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. stationarityassumption

14 Slide © 2006 Thomson/South-Western Binomial Distribution Our interest is in the number of successes Our interest is in the number of successes occurring in the n trials. occurring in the n trials. Our interest is in the number of successes Our interest is in the number of successes occurring in the n trials. occurring in the n trials. We let x denote the number of successes We let x denote the number of successes occurring in the n trials. occurring in the n trials. We let x denote the number of successes We let x denote the number of successes occurring in the n trials. occurring in the n trials.

15 Slide © 2006 Thomson/South-Western 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 Binomial Distribution n Binomial Probability Function

16 Slide © 2006 Thomson/South-Western Binomial Distribution n Binomial Probability 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

17 Slide © 2006 Thomson/South-Western Binomial Distribution n Example: 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 next year. 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.

18 Slide © 2006 Thomson/South-Western Binomial Distribution n 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? Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Let : p =.10, n = 3, x = 1

19 Slide © 2006 Thomson/South-Western n Tree Diagram Binomial Distribution 1 st Worker 2 nd Worker 3 rd Worker x x Prob. Leaves (.1) Leaves (.1) Stays (.9) Stays (.9) Leaves (.1) S (.9) Stays (.9) S (.9) L (.1)

20 Slide © 2006 Thomson/South-Western Binomial Distribution E ( x ) =  = np Var( x ) =  2 = np (1  p ) n Expected Value n Variance n Standard Deviation

21 Slide © 2006 Thomson/South-Western Binomial Distribution E ( x ) =  = 3(.1) =.3 employees out of 3 Var( x ) =  2 = 3(.1)(.9) =.27 n Expected Value n Variance n Standard Deviation