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Understanding Probability Density and Continuous Income in Business Calculus

This section explores key concepts in business calculus, focusing on probability density functions, continuous income streams, and consumer/producer surplus. Students will learn to construct and interpret probability density functions, evaluate continuous income streams, and calculate consumer and producer surplus. Essential formulas and examples illustrate the application of these concepts in real-world scenarios, such as predicting water usage and calculating total income over a designated period. By the end of this section, students will gain practical skills for analyzing business-related economic phenomena.

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Understanding Probability Density and Continuous Income in Business Calculus

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  1. Learning Objectives for Section 14.2 Applications in Business/Economics 1. The student will be able to construct and interpret probability density functions. 2, The student will be able to evaluate a continuous income stream. 3. The student will be able to evaluate consumers’ and producers’ surplus. Barnett/Ziegler/Byleen Business Calculus 11e

  2. Probability Density Functions • A probability density function must satisfy: • f (x)  0 for all x • The area under the graph of f (x) is 1 • If [c, d] is a subinterval then Probability (c x  d) = Barnett/Ziegler/Byleen Business Calculus 11e

  3. Probability Density Functions(continued) Sample probability density function Barnett/Ziegler/Byleen Business Calculus 11e

  4. Example 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. Barnett/Ziegler/Byleen Business Calculus 11e

  5. 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. Barnett/Ziegler/Byleen Business Calculus 11e

  6. 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 Barnett/Ziegler/Byleen Business Calculus 11e

  7. Example 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 Barnett/Ziegler/Byleen Business Calculus 11e

  8. Example 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 Barnett/Ziegler/Byleen Business Calculus 11e

  9. Future Valueof a Continuous Income Stream From previous work we are familiar with the continuous compound interest formula A = Pert. If f (t) is the rate of flow of a continuous income stream, 0  t  T, and if the income is continuously invested at a rate r compounded continuously, the the future value FV at the end of T years is given by Barnett/Ziegler/Byleen Business Calculus 11e

  10. Example Let’s continue the previous example where f (t) = 600 e0.06 t Find the future value in 2 years at a rate of 10%. Barnett/Ziegler/Byleen Business Calculus 11e

  11. Example Let’s continue the previous example where f (t) = 600 e0.06 t Find the future value in 2 years at a rate of 10%. r = 0.10, T = 2, f (t) = 600 e 0.06t Barnett/Ziegler/Byleen Business Calculus 11e

  12. CS x p Consumers’ Surplus If is a point on the graph of the price-demand equation P = D(x), the consumers’ surplus CS at a price level of is which is the area between p = and p = D(x) from x = 0 to x = The consumers’ surplus represents the total savings to consumers who are willing to pay more than for the product but are still able to buy the product for . Barnett/Ziegler/Byleen Business Calculus 11e

  13. Example Find the consumers’ surplus at a price level of for the price-demand equation p = D (x) = 200 – 0.02x Barnett/Ziegler/Byleen Business Calculus 11e

  14. Example Find the consumers’ surplus at a price level of for the price-demand equation p = D (x) = 200 – 0.02x Step 1. Find the demand when the price is Barnett/Ziegler/Byleen Business Calculus 11e

  15. Example (continued) Step 2. Find the consumers’ surplus: Barnett/Ziegler/Byleen Business Calculus 11e

  16. CS x p Producers’ Surplus If is a point on the graph of the price-supply equation p = S(x), then the producers’ surplus PS at a price level of is which is the area between and p = S(x) from x = 0 to The producers’ surplus represents the total gain to producers who are willing to supply units at a lower price than but are able to sell them at price . Barnett/Ziegler/Byleen Business Calculus 11e

  17. Example Find the producers’ surplus at a price level of for the price-supply equation p = S(x) = 15 + 0.1x + 0.003 2 Barnett/Ziegler/Byleen Business Calculus 11e

  18. Example Find the producers’ surplus at a price level of for the price-supply equation p = S(x) = 15 + 0.1x + 0.003x2 Step 1. Find , the supply when the price is Solving for using a graphing utility: Barnett/Ziegler/Byleen Business Calculus 11e

  19. Example (continued) Step 2. Find the producers’ surplus: Barnett/Ziegler/Byleen Business Calculus 11e

  20. Summary • We learned how to use a probability density function. • We defined and used a continuous income stream. • We found the future value of a continuous income stream. • We defined and calculated a consumer’s surplus. • We defined and calculated a producer’s surplus. Barnett/Ziegler/Byleen Business Calculus 11e

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