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FURTHER APPLICATIONS OF INTEGRATION

8. FURTHER APPLICATIONS OF INTEGRATION. FURTHER APPLICATIONS OF INTEGRATION. 8.5 Probability. In this section, we will learn about: The application of calculus to probability. PROBABILITY. Calculus plays a role in the analysis of random behavior. PROBABILITY.

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FURTHER APPLICATIONS OF INTEGRATION

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  1. 8 FURTHER APPLICATIONS OF INTEGRATION

  2. FURTHER APPLICATIONS OF INTEGRATION • 8.5 • Probability • In this section, we will learn about: • The application of calculus to probability.

  3. PROBABILITY • Calculus plays a role in the analysis of random behavior.

  4. PROBABILITY • Suppose we consider any of the following: • Cholesterol level of a person chosen at random from a certain age group • Height of an adult female chosen at random • Lifetime of a randomly chosen battery of a certain type

  5. CONTINUOUS RANDOM VARIABLES • Such quantities are called continuous random variables. • This is because their values actually range over an interval of real numbers—although they might be measured or recorded only to the nearest integer.

  6. PROBABILITY • Given the earlier instances, we might want to know the probability that: • A blood cholesterol level is greater than 250. • The height of an adult female is between 60 and 70 inches. • The battery we are buying lasts between 100 and 200 hours.

  7. PROBABILITY • If Xrepresents the lifetime of that type of battery, we denote this last probability as: P(100 ≤ X ≤ 200)

  8. PROBABILITY • According to the frequency interpretation of probability, that number is the long-run proportion of all batteries of the specified type whose lifetimes are between 100 and 200 hours. • As it represents a proportion, the probability naturally falls between 0 and 1.

  9. PROBABILITY DENSITY Equation 1 • Every continuous random variable X has a probability density functionf. • This means that the probability that X lies between a and b is found by integrating ffrom a to b:

  10. PROBABILITY DENSITY • Here is the graph of a model for the probability density function f for a random variable X. • X is defined to be the height in inches of an adult female in the United States.

  11. PROBABILITY DENSITY • The probability that the height of a woman chosen at random from this population is between 60 and 70 inches is equal to the area under the graph of f from 60 to 70.

  12. PROBABILITY DENSITY • In general, the probability density function f of a random variable X satisfies the condition f(x) ≥ 0 for all x.

  13. PROBABILITY DENSITY Equation 2 • As probabilities are measured on a scale from 0 to 1, it follows that:

  14. PROBABILITY DENSITY Example 1 • Let f(x) = 0.006x(10 – x) for 0 ≤ x ≤ 10 and f(x) = 0 for all other values of x. • a. Verify that f is a probability density function. • b. Find P(4 ≤ X ≤ 8).

  15. PROBABILITY DENSITY Example 1 a • For 0 ≤ x ≤10, we have • 0.006x(10 – x) ≤ 0. • So, f(x) ≥ 0 for all x.

  16. PROBABILITY DENSITY Example 1 a • We also need to check that Equation 2 is satisfied: • Hence, f is a probabilitydensity function.

  17. PROBABILITY DENSITY Example 1 b • The probability that X lies between 4 and 8 is:

  18. PROBABILITY DENSITY Example 2 • Phenomena such as waiting times and equipment failure times are commonly modeled by exponentially decreasing probability density functions. • Find the exact form of such a function.

  19. PROBABILITY DENSITY Example 2 • Think of the random variable as being the time you wait on hold before an agent of a company you’re telephoning answers your call. • So, instead of x, let’s use t to represent time, in minutes.

  20. PROBABILITY DENSITY Example 2 • If f is the probability density function and you call at time t = 0, then, from Definition 1, • The probability that an agent answers within the first two minutes is represented by: • The probability that your call is answered during the fifth minute is represented by:

  21. PROBABILITY DENSITY Example 2 • It’s clear that f(t) = 0 for t < 0 • The agent can’t answer before you place the call.

  22. PROBABILITY DENSITY Example 2 • For t > 0, we are told to use an exponentially decreasing function. • That is, a function of the form f(t) = Ae–c t, where A and c are positive constants. • Therefore,

  23. PROBABILITY DENSITY Example 2 • We use Equation 2 to find the value of A:

  24. PROBABILITY DENSITY Example 2 • Therefore, A/c = 1, and so A = c. • Thus, every exponential density function has the form

  25. PROBABILITY DENSITY • A typical graph is shown here.

  26. AVERAGE VALUES • Suppose you’re waiting for a company to answer your phone call—and you wonder how long, on average, you can expect to wait. • Let f(t) be the corresponding density function, where t is measured in minutes. • Then, think of a sample of N people who have called this company.

  27. AVERAGE VALUES • Most likely, none had to wait over an hour. • So, let’s restrict our attention to the interval 0 ≤ t ≤ 60. • Let’s divide that interval into n intervals of length Δt and endpoints 0, t1, t2, …, t60. • Think of Δt as lasting a minute, half a minute, 10 seconds, or even a second.

  28. AVERAGE VALUES • The probability that somebody’s call gets answered during the time period from ti–1 to ti is the area under the curve y = f(t) from ti–1 to ti. • This is approximately equal to f( ) Δt.

  29. AVERAGE VALUES • This is the area of the approximating rectangle in the figure, where is the midpoint of the interval.

  30. AVERAGE VALUES • The long-run proportion of calls that get answered in the time period from ti–1to ti is f( ) Δt.

  31. AVERAGE VALUES • So, out of our sample of N callers, we expect that: • The number whose call was answered in that time period is approximately Nf( ) Δt. • The time that each waited is about .

  32. AVERAGE VALUES • Therefore, the total time they waited is the product of these numbers: • approximately

  33. AVERAGE VALUES • Adding over all such intervals, we get the approximate total of everybody’s waiting times:

  34. AVERAGE VALUES • Dividing by the number of callers N, we get the approximate average waiting time: • We recognize this as a Riemann sum for the function tf(t).

  35. MEAN WAITING TIME • As the time interval shrinks (that is, Δt → 0 and n→ ∞), this Riemann sum approaches the integral • This integral is called the mean waiting time.

  36. MEAN • In general, the meanof any probability density function f is defined to be: • It is traditional to denote the mean by the Greek letter μ(mu).

  37. INTERPRETING MEAN • The mean can be interpreted as: • The long-run average value of the random variable X • A measure of centrality of the probability density function

  38. EXPRESSING MEAN • The expression for the mean resembles an integral we have seen before.

  39. EXPRESSING MEAN • Suppose R is the region that lies under the graph of f.

  40. EXPRESSING MEAN • Then,weknow from Formula 8 in Section 8.3 that the x-coordinate of the centroid of Ris: • This is because of Equation 2.

  41. EXPRESSING MEAN • Thus, a thin plate in the shape of R balances at a point on the vertical line x = µ.

  42. MEAN Example 3 • Find the mean of the exponential distribution of Example 2:

  43. MEAN Example 3 • According to the definition of a mean, we have:

  44. MEAN Example 3 • To evaluate that integral, we use integration by parts, with u = t and dv = ce–ct dt:

  45. MEAN Example 3 • The mean is µ = 1/c. • So,we can rewrite the probability density function as:

  46. AVERAGE VALUES Example 4 • Suppose the average waiting time for a customer’s call to be answered by a company representative is five minutes. • Find the probability that a call is answered during the first minute. • Find the probability that a customer waits more than five minutes to be answered.

  47. AVERAGE VALUES Example 4 a • We are given that the mean of the exponential distribution is µ = 5 min. • So, from the result of Example 3, we know that the probability density function is:

  48. AVERAGE VALUES Example 4 a • Thus, the probability that a call is answered during the first minute is: • About 18% of customers’ calls are answered during the first minute.

  49. AVERAGE VALUES Example 4 b • The probability that a customer waits more than five minutes is: • About 37% of customers wait more than five minutes before their calls are answered.

  50. AVERAGE VALUES • Notice the result of Example 4 b: • Though the mean waiting time is 5 minutes, only 37% of callers wait more than 5 minutes. • The reason is that some callers have to wait much longer (maybe 10 or 15 minutes), and this brings up the average.

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