2 Probability Distributions and Their Applications Discrete DistributionsBinomial distributionPoisson distributionContinuous DistributionsNormal distributionLognormal distributionGamma distributionPearson Type III distributionGumbel’s Extremal distribution
3 Binomial distribution Consider a discrete time scale.At each point on this time scale, an event may either occur or not occur.Let the probability of the event occurring be p for every point on the time scale.Thus, the occurrence of the event at any point on the time scale is independent of the history of any prior occurrences or non-occurrences.The probability of an occurrence at the i-th point on the time scale is p for i = 1,2,…...A process having these properties is said to be a Bernoulli process.
4 As an example of a Bernoulli process consider that during any year the probability of the maximum flow exceeding 10,000 cubic feet per second (cfs) on a particular river is p.Common terminology for a flow exceeding a given value is an exceedance.Further consider that the peak flow in any year is independent from year to year (a necessary condition for the process to be a Bernoulli process).Let be the probability of not exceeding 10,000 cfs.We can neglect the probability of a peak exactly cfs since the peak flow rates would be a continuous process so the probability of a peak exactly cfs would be zero.In this example, the time scale is discrete with the points nominally 1 year in time apart.We can now make certain probabilistic statements about the occurrence of a peak flow in excess of cfs (an exceedance).
5 For example, the probability of an exceedance occurring in year 3 and not in year 1 or 2 is since the process is independent from year to year.The probability of (exactly) one exceedance in any 3-year period is since the exceedance could occur in either the first, second, or third year.Thus the probability of (exactly) one exceedance in three years isIn a similar manner, the probability of 2 exceedances in 5 years can be found from the summation of the terms.It can be seen that each of theses terms is equivalent to and that the number of terms is equal to the number of ways of arranging 2 items (the p's) among 5 items (the p's and q's).Therefore the total number of terms is or 10 so that the probability of exactly 2 exceedances in 5 years is 10
6 This result can be generalized so that the probability of exceedances in n years is The result is applicable to any Bernoulli process so that the probability of occurrences of an event in n independent trials if p is the probability of an occurrence in a single trial is given by:This equation is known as the binomial distribution.The binomial distribution and the Bernoulli process are not limited to a time scale. Any process that may occur with probability p at discrete points in time or space or in individual trials may be a Bernoulli process and follow the binomial distribution
7 The cumulative binomial distribution is and gives the probability of or fewer occurrences of an event in n independent trials if the probability of an occurrence in any trial is p.Continuing the above example, the probability of less than 3 exceedances in 5 years isThe mean and variance of the binomial distribution are
8 The coefficient of skew is so that the distribution is symmetrical forskewed to the right forand skewed to the left forThe binomial distribution has an additive property. That is if X has a binomial distribution with parameters n1 and p1 and Y has a binomial distribution with parameters n2 and p2, then Z=X+Y has a binomial distribution with parameters n=n1+n2 and p.
9 Example: In order to be 90 percent sure that a design storm is not exceeded in a 10 year period. What should be the return period of the design storm?Solution: Let p be the probability of the design storm being exceeded. The probability of no exceedances is given by:years.
10 Poisson distributionThe Poisson distribution is like the binomial distribution in that it describes phenomena for which the average probability of an event is constant, independent of the number of previous events. In this case, however, the system undergoes transitions randomly from one state with n occurrences of an event to another with (n+1) occurrences, in a process that is irreversible. That is, the ordering of the events cannot be interchanged. Another distinction between the binomial and Poisson distributions is that for the Poisson process the number of possible events should be large.The Poisson distribution may be inferred from the identitywhere the most probable number of occurrences of the event is .
11 If the factorial is expanded in a power series expansion, the probability P(r) that exactly r random occurrences will take place can be inferred as the th term in the series, i.e.,This probability distribution leads directly to the interpretation that:= the probability that an event will not occur,= the probability that an event will occur exactly once,= the probability that an event will occur exactly twice, etc,
12 The mean and the variance of the Poisson distribution are: The coefficient of skew is so that as gets large, the distribution goes from a positively skewed distribution to a nearly symmetrical distribution. The cumulative Poisson probability that an event will occur x times or less is:Of course, the probability that the event will occur (x+1) or more times would be the complement of P(x).
13 Example: what is the probability that a storm with a return period of 20 years with occur once in 10-year period ?Solution:
14 Compare bi-nomial with Poissions The binomial distribution is useful for systems with two possible outcomes of events (failure–no failure) in cases where there is a known, finite number of (Bernoulli) trials and the ordering of the trials does not affect the outcome.The Poisson distribution treats systems in which randomly occurring phenomena cause irreversible transitions from one state to another.