Bush__The_Essential_Physics_for_Medical_Imaging - Biomedical ...

Toss of a coinToss of a dieObservation of a radioactivenucleus for a time "t"Observation of a detector ofefficiency E placed near aradioactive nucleus for a time "t""Heads""A four"It decays1/21/61 - e-AtSource: Adapted from Knoll, GF.Radiation detection and measurement, 3rd ed. New York: John Wiley,2000.Probability Distribution Functions-Binomial, Poisson,and GaussianA probability distribution function (pdf) (also known as the probability densityfuncton) describes the probability of obtaining each outcome from a measurement-forexample, the probability of obtaining six "heads" in a throw of tencoins. There are three pdfs relevant to binary processes-the binomial, the Poisson,and the Gaussian (normal). The binomial pdf exactly describes the probabilityof each outcome from a measurement of a binary process:P(x) = '(:~ )' YO - p)N-xx. x.where N is the total number of trials in a measurement, p is the probability of successin a single trial, and x is the number of successes. The mathematical notationN!, called ftctorial notation, is simply shorthand for the productFor example, 5! = 5·4·3·2·1 = 120. If we wish to know the probability of obtainingtwo heads in a toss of four coins, x = 2, N = 4, and p = 0.5. The probability ofobtaining two heads isP(two-heads) = 4! (0.5F 0 - 0.5)4-2 = 0.3752!(4 - 2)!Figure 20-28 is a graph of the binomial pd£ It can be shown that the sum ofthe probabilities of all outcomes for the binomial pdf is 1.0 and that the mean (X)and standard deviation (cr)of the binomial pdf are as follows:x = pN and cr = YpN(J - p) [20-12]If the probability of a success in a trial is much less than 1 (not true for a tossof a coin, but true for most radiation measurements), the standard deviation isapproximated by the following:cr = VpN(J - p) = yPN = ~ [20-13]Because of the factorials in Equation 20-10, it is difficult to use if either x or Nis large. The Poisson and Gaussian pdfs are approximations to the binomial pdf thatare otten used when x or N is large. Figure 20-29 shows a Gaussian pd£