Download presentation
Presentation is loading. Please wait.
Published byEdmund Hart Modified over 9 years ago
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”
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.