Survey

Document related concepts

no text concepts found

Transcript

236 Chapter 4 Special Distributions Proof Suppose an event has occurred at time a. Consider the interval that extends from a to a + y. Since the (Poisson) events are occurring at the rate of Î» per unit time, âÎ»y 0 the probability that no outcomes will occur in the interval (a, a + y) is e 0!(Î»y) = eâÎ»y . Deï¬ne the random variable Y to denote the interval between consecutive occurrences. Notice that there will be no occurrences in the interval (a, a + y) only if Y > y. Therefore, P(Y > y) = eâÎ»y or, equivalently, FY (y) = P(Y â¤ y) = 1 â P(Y > y) = 1 â eâÎ»y Let f Y (y) be the (unknown) pdf for Y . It must be true that % y P(Y â¤ y) = f Y (t) dt 0 Taking derivatives of the two expressions for FY (y) gives % y d d (1 â eâÎ»y ) f Y (t) dt = dy 0 dy which implies that f Y (y) = Î»eâÎ»y , y >0 Case Study 4.2.4 Over âshortâ geological periods, a volcanoâs eruptions are believed to be Poisson eventsâthat is, they are thought to occur independently and at a constant rate. If so, the pdf describing the intervals between eruptions should have the form f Y (y) = Î»eâÎ»y . Collected for the purpose of testing that presumption are the data in Table 4.2.5, showing the intervals (in months) that elapsed between thirty-seven consecutive eruptions of Mauna Loa, a fourteenthousand-foot volcano in Hawaii (106). During the period coveredâ1832 to 1950âeruptions were occurring at the rate of Î» = 0.027 per month (or once every 3.1 years). Is the variability in these thirty-six yi âs consistent with the statement of Theorem 4.2.3? Table 4.2.5 126 73 26 6 41 26 73 23 21 18 11 3 3 2 6 6 12 38 6 65 68 41 38 50 37 94 16 40 77 91 23 51 20 18 61 12 To answer that question requires that the data be reduced to a densityscaled histogram and superimposed on a graph of the predicted exponential pdf (Continued on next page)