.32. plotted in Figure 1.8. The X axis gives the event (for example telephone line failure) and the y axis the probability of the event p(m)=" (1.32) occurring. The curve shows that most of the events occur close to the mean value and this is usually the case in nature. The normal curve is given by Equation 1.38, where x is the mean of the values The binomial distribution is used for discrete events and is applic- making up the curve and o is their standard deviation. able if the probability of occurrence p of an event is constant on each trial. The mean of the distribution B(M) and the standard deviation B(S) are given by Equations 1.33 and 1.34. B(M) np (1.33) B(S)=(npq)" = (1.34) 1.9.2 Poisson distribution The Poisson distribution is used for discrete events and, like the Telephone 68.26% line failure binomial distribution, it applies to mutually independent events. It is used in cases where pand q cannot both be defined. For example, 95.46% one can state the number of times a telephone circuit failed over a 99.73% given period of time, but not the number of times when it did not fail. The Poisson distribution may be considered to be the limiting case Figure 1.8 The normal curve of the binomial when n is large and p is small. The probability of an event occurring m successive times is given by the Poisson distribu- tion as in Equation 1.35. (1.38) (m) = (np) m e " = (1.35) Different distributions will have varying mean and standard devi- ations but if they are distributed normally their curves will all follow The mean P(M) and standard deviation P(S) of the Poisson dis- Equation 1.38. These distributions can be normalised to a standard tribution are given by Equations 1.36 and 1.37. form by moving the origin of their normal curve to their mean value, shown as B in Figure 1.8. The deviation from the mean is now P(M) = n p = (1.36) represented on a new scale of units given by Equation 1.39. P(S)=(np)" (1.37) (1.39) Poisson probability calculations can be done by the use of prob- ability charts as shown in Figure