Validity and application of some continuous distributions Dr. Md. Monsur Rahman Professor Department of Statistics University of Rajshahi Rajshshi –

Normal distribution The first discoverer of the normal probability function was Abraham De Moivre( ), who, in 1733, derived the distribution as the limiting form of the binomial distribution. But the same formula was derived by Karl Freidrich Gauss( ) in connection with his work in evaluating errors of observation in astronomy. This is why the normal probability is often referred to as Gaussian distribution. 2

X: Normal Variate Density: Standard Normal Variate : 3

Normal distribution 4

Properties of Normal distribution Normal probability curve is symmetrical about the ordinate at Mean, median and mode of the distribution are equal and each of these is The curve has its points of inflection at By a point of infection, we mean a point at which the concavity changes All odd order moments of the distribution about the mean vanish The values of and are 0 and 3 respectively 5

includes about 68.27% of the population includes about 95.45% of the population includes about 99.73% of the population Application: Many biological characteristics conform to a Normal distribution - for example, heights of adult men and women, blood pressures in a healthy population, RBS levels in blood etc. 6

Validity of Normal Distribution for a set of data Many statistical methods can only be used if the observations follow a Normal Distribution. There are several ways of investing whether observations follow a Normal distribution. With a large sample we can inspect a histogram to see whether it looks like a Normal distribution curve. This does not work well with a small sample, and a more reliable method is the normal plot which is described below. 7

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X: Normal Variate Density: Standard Normal Variate : 9

CDF OF X : F(X) CDF OF Z : P quantile of X : P quantile of Z :, is the solution of 10

Dataset Find empirical CDF values Arrange the data in ascending order as Empirical CDF values are as follows Using normal table obtain the values corresponding to 11

If the given set of observations follow normal distribution, the plot (x, z) should roughly be a straight line and the line passes through the point and has slope. Graphical estimates of and may be obtained. If the data are not come from Normal distribution we will get a curve of some sort. 12

Table 1 : RBS levels(mmol/L) measured in the blood of 20 medical students. Data of Bland(1995), pp. 66 Bland,M.(1995): An Introductions to Medical Statistics, second edition, ELBS with Oxford University Press. 13

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MLE 15

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Goodness of Fit Test We use here Kolmogorov-Smirnov (KS) test for the given data KS statistic=max |CDF_FIT- CDF_EMP| For the RBS level data we calculate KS statistic KS(cal)= % tabulated value=0.294 Conclusion: Normal distribution fit is good for the given data 17

Estimated population having RBS within the normal range (3.9 – 7.8mmol/L) is about 51% Estimated population having RBS below the normal range is about 49% Estimated population having RBS above the normal range is 0% Results 18

Empirical CDF values of are as follows: Obtain the values corresponding to Similarly values are obtained corresponding to Two sample case 19

If the first set of data come from normal distribution with mean and variance, then the plot will roughly be linear and passes through the point with slope. If the second set of data come from normal distribution with mean and variance, then the plot will roughly be linear and passes through the point with slope. 20

Both the lines parallel indicating different means but equal variances Both the lines coincide indicating equal means and equal variances Both the lines pass through the same point on the X-axis indicating same means but different variances 21

Table 2 : Burning times (rounded to the nearest tenth of a minute) of two kinds of emergency flares. Data due to Freund and Walpole(1987), pp. 530 Brand A: 14.9,11.3,13.2,16.6,17.0,14.1,15.4, 13.0,16.9 Brand B: 15.2,19.8,14.7,18.3,16.2,21.2,18.9, 12.2,15.3,19.4 Freund, J.E. and Walpole, R.E.(1987): Mathematical Statistics, Fourth edition, Prentice-Hall Inc. 22

Above plot indicates that both the samples come from normal population with unequal means and variances 23

Log-normal distribution In probability theory, a log-normal distribution is aprobability theory probability distributionprobability distribution of a random variable whoserandom variable logarithmlogarithm is normally distributed. If X is a randomnormally distributed variable with a normal distribution, then Y = exp(X) hasexp a log-normal distribution; likewise, if Y is log-normally distributed, then X = log(Y) is normally distributed. It is occasionally referred to as the Galton distribution. 24

Density: Mean = Variance= Median= Mode= 25

Log-normal density function x f(x) 26

Application Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations), vitamin D level in blood etc. follow lognormal distribution. Subsequently, reference ranges for measurements inreference ranges healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean. 27

Table 3 : Vitamin D levels(ng/ml) measured in the blood of 26 healthy men. Data due to Bland(1995), pp Bland,M.(1995): An Introductions to Medical Statistics, Second edition, ELBS with Oxford University Press. 28

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ng/ml MLE 30

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Goodness-of-fit test KS statistic=max |CDF_FIT- CDF_EMP| For the vitamin D level data we calculate KS statistic KS(cal)= % tabulated value=0.274 Conclusion: Lognormal distribution fit is good for the given vitamin D data 32

Estimated population having vitamin D level within the normal range (30 – 74 ng/ml) is about 56% Estimated population having vitamin D level below the normal range is about 40% Estimated population having vitamin D level above the normal range is about 4% Results 33

Weibull Distribution Weibull distribution is used to analyze the lifetime data T: Lifetime variable Density function : Scale parameter(.632 quantile) : Shape parameter( 1 or =1) CDF : Reliability (or Survival) function: 34

Increasing hazard rate : for Decreasing hazard rate: for Constant hazard rate : for p quantile, which is the solution of Accordingly, Hazard Function : 35

Density function: : Scale parameter(.632 quantile) CDF : Reliability (or Survival) function: Weibull distribution reduces to exponential distribution when Exponential distribution 36

p quantile, which is the solution of Accordingly, Hazard Function : 37

The red curve is the exponential density The red line is the exp. hazard function 38

From the Weibull CDF we get where Ordered lifetimes are: values are obtained through the empirical CDF values as given below Validity of Weibull distribution for a set of data 39

If the data follow Weibull distribution with scale parameter and shape parameter, the plot of (X,Y) will roughly be linear with slope and passes through the point. Accordingly, the graphical estimates of and may be obtained. 40

Table 4: Specimens lives (in hours) of a electrical insulation at temperature appear below. Data due to Nelson(1990), pp. 154 Nelson,W.(1990): Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, John Wiley and Sons. 2520, 2856, 3192, 3192,

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MLE of and Log-likelihood function of and based on observed data MLE of and by maximizing the log-Likelihood with respect to and using numerical method. Graphical estimates may be used as starting values required for the numerical method The MLEs of and are denoted by and respectively. 43

For the insulation fluid data given in table 4 the following results (based on MLEs) are obtained: hours Estimated median life= hours ML estimate of R(t) Time (hour): Reliability :

Weibull versus Exponential Model Suppose we want to test whether we accept exponential or Weibull model for a given set of data The above test is equivalent to test whether the shape parameter of Weibull distribution is unity or not i.e. vs 45

Test Procedure(LR test) Under the log-likelihood function is which yields, MLE of. Maximum of is given by 46

Similarly, under the maximum of the log-likelihood is given by where and are the MLE s of and under. LR test implies follows chi-square distribution with 1 df. If, accept (use) exponential Model 47

If, accept (use) Weibull model For the insulation fluid data given in table 4 Conclusion: Weibull model may be accepted at 5% level of significance 48

Accelerated Life Testing (ALT) for Weibull Distribution Stress: Temperature, Voltage, Load, etc. Under operating (used) stress level, it takes a lot of time to get sufficient number of failures Lifetimes obtained under high stress levels Aim: (i) To estimate the lifetime distribution under used stress level, say, (ii) To estimate reliability for a specified time under (iii) To estimate quantiles under

Sampling scheme(under constant stress testing) Divide n components into k groups with number of components respectively, where components exposed under stress levels, j-th lifetime corresponding to Obtain the equation for the lifetime corresponding to i-th group 50

If the data corresponding to the i- th group follow, the plot will roughly be linear with slope and passes through the point If the plots are linear and parallel, then lifetimes under different stress levels are Weibull with common slope and different scale which implies that depends on the stress levels The equation for the lifetime corresponding to i-th group 51

If the k plots are linear and parallel, the lifetimes under different stress levels are Weibull with different slopes and different scales which implies that both and both depend on the stress levels. In this case modeling is difficult. For the first case the relationship between the life and stress will be identified Plot log(.632 quantile) against the stress levels If the plot yields a straight line then the life-stress relationship will be 52

Estimation of Likelihood function under stress level where, Total log-likelihood using ML method 53

Using numerical method MLEs of may be obtained MLE of at, say,, is obtained through the relationship Hence ML estimate of Weibull density under used stress level is obtained. Accordingly, estimate of reliability for a specified time, median life and other desired percentiles may also be obtained 54

Table 5: Specimens lives (in hours) of a electrical insulation at three temperatures appear below, data of Nelson(1990), pp Nelson,W.(1990): Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, John Wiley and Sons. 55

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Above three plots of the data given in table5 are roughly linear and parallel, so the lifetimes under three stress levels are Weibull with common slope and different scale parameters which implies that the scale parameters depend on the stress levels Arrhenious life-stress relationship (temperature stress) where W is the temperature in degree kelvin Temperature in degree kelvin= temperature in degree centigrade plus

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Results based on MLEs for the data given in table 5 with respect to the Arrhenious-Weibull model At used stress(180 deg. Centigrade) the following results are obtained Estimated median lifetime= hours and 59

At used stress(180 deg. Centigrade) the following results are obtained Estimated median lifetime= hours Results based on MLEs for the data given in table 5 with respect to the Arrhenious-Exponential model 60

Weibull versus Exponential Model for ALT vs (For Exponential model) (For Weibull model) Conclusion: Accept Weibull model at 5% level of significance 61