1. 1 1 Slide © 2016 Cengage Learning. All Rights Reserved. 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. Chapter 5 - Discrete Probability Distributions

2. 2 2 Slide © 2016 Cengage Learning. All Rights Reserved. Random Variables… A random variable is a function or rule that assigns a number to each outcome of an experiment. Basically it is just a symbol that represents the outcome of an experiment. X = number of heads when the experiment is flipping a coin 20 times. C = the daily change in a stock price. R = the number of miles per gallon you get on your auto during the drive to your family’s home. Y = the amount of sugar in a mountain dew (not diet of course). V = the speed of an auto registered on a radar detector used on I-83

3. 3 3 Slide © 2016 Cengage Learning. All Rights Reserved. Two Types of Random Variables… Discrete Random Variable – usually count data [Number of] * one that takes on a countable number of values – this means you can sit down and list all possible outcomes without missing any, although it might take you an infinite amount of time. X = values on the roll of two dice: X has to be either 2, 3, 4, …, or 12. Y = number of customer at Starbucks during the day Y has to be 0, 1, 2, 3, 4, 5, 6, 7, 8, ……………”real big number” Continuous Random Variable – usually measurement data [time, weight, distance, etc] * one that takes on an uncountable number of values – this means you can never list all possible outcomes even if you had an infinite amount of time. X = time it takes you to walk home from class: X > 0, might be 5.1 minutes measured to the nearest tenth but in reality the actual time is 5.10000001…………………. minutes?)

4. 4 4 Slide © 2016 Cengage Learning. All Rights Reserved. Random Variables Question Random Variable x Type Familysize x = Number of dependents reported on tax return reported on tax return Discrete 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

5. 5 5 Slide © 2016 Cengage Learning. All Rights Reserved. We can describe a discrete probability distribution with a table, We can describe a discrete probability distribution with a table, graph, or formula. graph, or formula. We can describe a discrete probability distribution with a table, We can describe a discrete probability distribution with a table, graph, or formula. graph, or formula. Discrete Probability Distributions The probability distribution is defined by a probability function, The probability distribution is defined by a probability function, Denoted by f ( x ), which provides the probability for each value Denoted by f ( x ), which provides the probability for each value of the random variable. of the random variable. The probability distribution is defined by a probability function, The probability distribution is defined by a probability function, Denoted by f ( x ), which provides the probability for each value Denoted by f ( x ), which provides the probability for each value of the random variable. of the random variable. The required conditions for a discrete probability function are: The required conditions for a discrete probability function are: f ( x ) > 0  f ( x ) = 1 The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

6. 6 6 Slide © 2016 Cengage Learning. All Rights Reserved. Discrete Uniform Probability Distribution The discrete uniform probability distribution is the The discrete uniform probability distribution is the simplest example of a discrete probability simplest example of a discrete probability distribution given by a formula. distribution given by a formula. The discrete uniform probability distribution is the The discrete uniform probability distribution is the simplest example of a discrete probability simplest example of a discrete probability distribution given by a formula. distribution given by a formula. 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 Example would be tossing a coin (Head or Tails) and f (x) = (1/n) =1/2 or rolling a die where f (x) = (1/n) = 1/6 the values of the random variable random variable are equally likely are equally likely

7. 7 7 Slide © 2016 Cengage Learning. All Rights Reserved. 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 Discrete Random Variable with a Finite Number of Values We can count the TVs sold, and there is a finite upper limit on the number that might be sold (the number of TVs in stock). Discrete Random Variable with an Infinite Sequence of Values 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,... We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

8. 8 8 Slide © 2016 Cengage Learning. All Rights Reserved. a tabular representation of the probability a tabular representation of the probability distribution for TV sales was developed. distribution for TV sales was developed. Using past data on TV sales, …Using past data on TV sales, … Number Number Units Sold of Days Units Sold of Days 0 80 1 50 1 50 2 40 2 40 3 10 3 10 4 20 4 20 200 200 x f ( x ) x f ( x ) 0.40 0.40 1.25 1.25 2.20 2.20 3.05 3.05 4.10 4.10 1.00 1.00 80/200 Discrete Probability Distributions Example: JSL Appliances

9. 9 9 Slide © 2016 Cengage Learning. All Rights Reserved. Expected Value The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. E ( x ) =  =  xf ( x ) E ( x ) =  =  xf ( x ) The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. E ( x ) =  =  xf ( x ) E ( x ) =  =  xf ( x ) The expected value is a weighted average of the The expected value is a weighted average of the values the random variable may assume. The values the random variable may assume. The weights are the probabilities. weights are the probabilities. The expected value is a weighted average of the The expected value is a weighted average of the values the random variable may assume. The values the random variable may assume. The weights are the probabilities. weights are the probabilities. The expected value does not have to be a value the The expected value does not have to be a value the random variable can assume. random variable can assume. The expected value does not have to be a value the The expected value does not have to be a value the random variable can assume. random variable can assume.

10. 10 Slide © 2016 Cengage Learning. All Rights Reserved. expected number of TVs sold in a day x f ( x ) xf ( x ) x f ( x ) xf ( x ) 0.40.00 0.40.00 1.25.25 1.25.25 2.20.40 2.20.40 3.05.15 3.05.15 4.10.40 4.10.40 E ( x ) = 1.20 E ( x ) = 1.20 Expected Value n Example: JSL Appliances

11. 11 Slide © 2016 Cengage Learning. All Rights Reserved. Variance and Standard Deviation 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 variance is a weighted average of the squared The variance is a weighted average of the squared deviations of a random variable from its mean. The deviations of a random variable from its mean. The weights are the probabilities. weights are the probabilities. The variance is a weighted average of the squared The variance is a weighted average of the squared deviations of a random variable from its mean. The deviations of a random variable from its mean. The weights are the probabilities. weights are the probabilities. Var( x ) =  2 =  ( x -  ) 2 f ( x ) 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.

12. 12 Slide © 2016 Cengage Learning. All Rights Reserved. 01234 -1.2-0.2 0.8 0.8 1.8 1.8 2.8 2.81.440.040.643.247.84.40.25.20.05.10.576.010.128.162.784 x -  ( x -  ) 2 f(x)f(x)f(x)f(x) ( x -  ) 2 f ( x ) Variance of daily sales =  2 = 1.660 x TVssquared Standard deviation of daily sales = 1.2884 TVs Variance n Example: JSL Appliances