2 Probability Distributions and Their Applications Continuous DistributionsNormal distributionExponential distributionGamma distributionLognormal distributionExtreme value distributionExtreme value type-I: Gumbel distributionExtreme value type-III (minimum): Weibull distributionPearson Type III distribution
3 Table: Area under standardized normal distribution
4 Central Limit theoremIf Sn is the sum of n independently and identically distributed random variables Xi each having a mean and variance then in the limit as n approaches infinitely, the distribution of Sn approaches a normal distribution with mean n and variance n
5 Constructing normal curves for data Frequently the histogram of a set observed data suggests that the data may be approximated by a normal distribution. One way to investigate the goodness of this approximation is by superimposing a normal curve on the frequency histogram and then visually compare the two distributions. Statistical procedures for testing the hypothesis that a set of data can be approximated by a normal (or any other) distribution.
6 Normal Approximations Bi-nomial DistributionIf X is a binomial random variable with parameters n1 and p and Y is a binomial random variable with parameters n2 and p the Z=X+Y is a binomial random variable with parameters n=n1+n2.Central Limit theorem would indicate that the normal distribution approximates the binomial distribution if n is large. Thus as n gets large the distribution ofApproaches a N(0,1). This is sometimes knows as the DeMoivre-Laplace limit theorem
7 Example-1: [Haan, 1979]X is a binomial random variable with n=25 and p=0.3. Compare the binomial and normal approximation to the binomial for evaluating the prob(5<X≤8).
8 Normal Approximations Poission DistributionThe sum of two Poissions random variables with parameters 1 and 2 is also a Poissions random variable with parameter =1+2 .. Extending this to the sum of a large number of Poission random variables, the Central Limit Theorem indicates that for large , the Poission may be approximated by a normal distribution. In this case the distribution ofapproaches a N(0,1). Since the Poission is the limiting form of the binomial and the binomial can be approximated by the normal, it is no surprise that the Poission can also be approximated by the normal.
9 Assignment-2:[due next week] X is a Poission random variable with =np where n=25 and p=0.3. Compare the Poission and normal approximation to the Poission for evaluating the prob(5<X≤8).
10 Exponential Distribution The exponential density function is given byand the cumulative exponential byThe mean and variance of the exponential distribution areThe exponential distribution is positively skewed with the skewness coefficient of 2. Both the method of moments and the maximum likelihood estimation give the parameter. The exponential distribution is a special case of the gamma distribution.
11 Example-2: [Haan, 1979]Haan and Johnson (1967) studied the physical characteristics of depressions in north-central Iowa. The data tabulated below shows the number of depressions falling into various classes based on the surface area of the depression. Plot a relative frequency histogram of the data. Superimpose on the histogram the best fitting exponential distribution. Estimate the probability that a depression selected at random will have an area greater than 2.25 acres.
12 Area (acres)No. depressions0-½1064-4½4½-1364½-551-1½185-5½21½-295½-662-2½126-6½32½-36½-713-3½7-7½3½-4
13 Gamma DistributionThe gamma distribution failure probability density obeys the equation, (4.6.1)where parameter r need not be an integer. The two parameters are the shape parameter r and the scale parameter The shape of the distribution depends significantly upon the value of r. It has also an impact on the hazard rate In the special case that r is an integer, the Erlangian distribution is recovered; in the special case that = 0.5 and r = 0.5, where is the number of degrees of freedom, the gamma distribution becomes the chi-square distribution.
14 The cumulative failure probability F(t) is: (4.6.2)The mean and variance of the gamma distribution are:andThe gamma distribution is especially appropriate for systems subjected to an environment of repetitive, random shocks generated according to the Poisson distribution; thus the failure probability depends upon how many shocks the device has suffered, i.e., its age. As another application, if the mean rate of wear of a device is a constant, but the rate of wear is subject to random variations, then the gamma function should be used.
15 For some devices, such as those for which corrosion of metals is important, it may be appropriate to modify the two-parameter gamma distribution by introducing a time delay before the onset of failures begins. Then equation (4.6.1) is modified to read as:(4.6.3)In such a case, the mean of the distribution becomes:(4.6.4)
16 Example-3:[MacCormick,1981] Suppose that a device subjected to repetitive random shocks satisfies a gamma distribution with parameters r = 3 and hr, and that no failures can occur until 200 hour have passed. Estimate (a) the probability of failure after the device has operated for t = 4500 hour and (b) its mean time to failure. (MacCormick, 1981, p. 37)Solution: In this problem, the time displacement is = 200 hour. Integrating equation (4.6.3) from 0 to t and using equation (4.6.2) gives the cumulative probability:Using equation (4.6.4) gives mean time to failure (MTTF) = /1 0-3 = 3200 hr.
17 Example-4: [Haan, 1979]The annual water yield for Cave Creek (near Fort Spring, Kentucky) is shown in the following table. Estimate the parameters of the gamma distribution for this data using both the method of moments and the method of maximum likelihood. Assuming the data follows a gamma distribution, estimate the probability of an annual water yield exceeding 20.0 inch.
18 Log-Normal Distribution The lognormal distribution (sometimes spelled out as the logarithmic normal distribution) of a random variable is one for which the logarithm of follows a normal or Gaussian distribution. Denote , then Y has a normal or Gaussian distribution given by:
19 Derived distribution: Since , , the distribution of can be found as: (4.13.2)Note that equation (4.13.1) gives the distribution of Y as a normal distribution with mean and variance Equation (4.13.2) gives the distribution of X as the lognormal distribution with parametersand
20 Estimation of parameters ( , ) of lognormal distribution: Note: , , Chow (1954) Method:(1)(2)(3)The mean and variance of the lognormal distribution are:The coefficient of variation of the Xs is:The coefficient of skew of the Xs is:Thus the lognormal distribution is skewed to the right; the skewness increasing with increasing values ofand
21 Example-5:[Haan, 1979, p. 78]Use the lognormal distribution and calculate the expected relative frequency for the third class interval on the data in table 5.1
22 Example-6: [Haan, 1979]Assume the data of table 5.1 follow the lognormal distribution. Calculate the magnitude of the 100-year peak flood.