1. Chapter 7 Additional Integration Topics Section 2 Applications in Business and Economics

2. 2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 7.2 Applications in Business/Economics The student will be able to: 1.Construct and interpret probability density functions. 2.Evaluate a continuous income stream. 3.Evaluate the future value of a continuous income stream. 4.Evaluate consumers’ and producers’ surplus.

3. 3 Random Variables  Random variables come in two varieties:  Discrete Values are distinct or separate and can be counted and listed  Continuous Infinite number of values that are within an interval Barnett/Ziegler/Byleen Business Calculus 12e

4. 4 Continuous vs Discrete  Discrete Random Variable Suppose we roll a single die. How many possible outcomes are there? There are 6 discrete possible outcomes.  Continuous Random Variable Suppose we randomly choose a real number (x) in the interval [1, 6]. How many possible outcomes are there? There are an infinite number of possible outcomes. Barnett/Ziegler/Byleen Business Calculus 12e

5. 5 Continuous Random Variables  Suppose an experiment is designed in such a way that any real number x on the interval [c, d] is a possible outcome.  Examples of what x could represent: Inches of rain in one day Height of a person between 5 ft and 7 ft Life of a lightbulb between 40 hours and 100 hours  These are all examples of continuous random variables because the possible outcomes are not discrete. Rather, there is an infinite number of possible outcomes over a specified interval. Barnett/Ziegler/Byleen Business Calculus 12e

6. 6 Probability Density Function Barnett/Ziegler/Byleen Business Calculus 12e

7. 7 Probability Density Functions A probability density function must satisfy 3 conditions: 1. f (x)  0 for all real x 2.The area under the graph of f (x) over the interval (- ,  ) is 1 3.If [c, d] is a subinterval of (- ,  ) then the probability that x falls in the interval [c, d] is equal to:

8. 8 Barnett/Ziegler/Byleen Business Calculus 12e Graph Examples

9. 9 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 In a certain city, the daily use of water in hundreds of gallons per household is a continuous random variable with probability density function Find the probability that a household chosen at random will use between 300 and 600 gallons. (Use graphing calculator.) There is a 23% probability that a household chosen at random uses between 300-600 gallons of water.

10. 10 Barnett/Ziegler/Byleen Business Calculus 12e Conceptual Insight The probability that a household in the previous example uses exactly 300 gallons is given by: In fact, for any continuous random variable x with probability density function f (x), the probability that x is exactly equal to a constant c is equal to 0. See khanAcademy.org “probability density functions” for an additional explanation.

11. 11 Example 2  Suppose that the length of a phone call (in minutes) is a continuous random variable with the probability density function:  Find the probability that a call selected at random will last 4 minutes or less. (Use graphing calculator.)  Solve for b so that the probability of a call selected at random lasting b minutes or less is 90%. Barnett/Ziegler/Byleen Business Calculus 12e

12. 12 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e There is a 63% probability that a phone call chosen at random will last 4 minutes or less. Find the probability that a call selected at random will last 4 minutes or less.

13. 13 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Solve for b so that the probability of a call selected at random lasting b minutes or less is 90%. There is a 90% probability of a call lasting 9.21 minutes or less.

14. 14 Application: Continuous Income  A function that models the flow of money represents a continuous income stream.  Let f(t) represent the rate of flow of a continuous income stream where t is time.  We can use calculus to find the total income produced over a specified time interval. Barnett/Ziegler/Byleen Business Calculus 12e

15. 15 Barnett/Ziegler/Byleen Business Calculus 12e Continuous Income Stream Total Income for a Continuous Income Stream: If f (t) is the rate of flow of a continuous income stream, the total income produced during the time period from t = a to t = b is a Total Income b

16. 16 Continuous Income Stream  This makes sense if you recall what we have been saying about definite integrals.  If you integrate a rate of change of a quantity on an interval then you get the total change of the quantity on that interval.  Since the rate of flow represents the rate of change of income produced then the definite integral from a to b represents the total income produced on that interval. Barnett/Ziegler/Byleen Business Calculus 12e

17. 17 Barnett/Ziegler/Byleen Business Calculus 12e Example 3 Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is f (t) = 600 e 0.06t

18. 18 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e The total income after the first 2 years is $1,274.97

19. 19 Example 3 (continued)  What would be the total income produced during the second two years? (Use graphing calculator.)  Interval will be [2, 4] because it represents the end of the 2 nd year to the end of the 4 th year. Barnett/Ziegler/Byleen Business Calculus 12e The total income produced during the next two years is $1437.52

20. 20 Homework #7-2A Pg 430 (13-19 odd, 21, 25) Barnett/Ziegler/Byleen Business Calculus 12e khanAcademy.org “discrete and continuous random variables” “probability density functions”