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Probability distribution functionsNormal distributionLognormal distributionMean, median and modeTailsExtreme value distributions

Normal (Gaussian) distributionNormal density function

What does the figure tell us about the values of the CDF?

More on the normal distributionP = normcdf(X,MU,SIGMA) returns the cdf of the normal distribution with mean MU and standard deviation SIGMA, evaluated at the values in X. The size of P is the common size of X, MU and SIGMA.normcdf(1)=0.8413. 1-normcdf(6)= 9.8659e-010If X is normally distributed, Y=aX+b is also normally distributed. What would be the mean and standard deviation of Y?Notation

Estimating mean and standard deviationGiven a sample from a normally distributed variable, the sample mean is the best linear unbiased estimator of the true mean.For the variance the equation gives the best unbiased estimator, but the square root is not an unbiased estimate of the standard deviation

x=randn(5,10000); s=std(x);mean(s) 0.9463s2=s.^2; mean(s2) 1.0106

Lognormal distributionIf ln(X) has normal distribution X has lognormal distribution. That is, if X is normally distributed exp(X) is lognormally distributed.Notation: Probability distribution function (PDF)

Mean and variance

Mean, mode and medianMode (highest point)Median (50% of samples)

Light and heavy tailsNormal distribution has light tail. Six sigma is equivalent to .999999999 (nine nines) safety.Lognormal is heavy tailed 0.9963m=exp(0.5)m =1.6487 v=exp(1)*(exp(1)-1)v =4.6708sig=sqrt(v)sig =2.1612sig6=m+6*sigsig6 =14.6159logncdf(sig6,0,1) =0.9963

Fitting distribution to dataTypically fit to CDF.

Empirical CDF[F,X] = ecdf(Y) calculates the Kaplan-Meier estimate of the cumulative distribution function (cdf), also known as the empirical cdf. Y is a vector of data values. F is a vector of values of the empirical cdf evaluated at X. [F,X,FLO,FUP] = ecdf(Y) also returns lower and upper confidence bounds for the cdf. These bounds are calculated using Greenwood's formula, and are not simultaneous confidence bounds. ecdf(...) without output arguments produces a plot of the empirical cdf. Use the data cursor to read precise values from the plot.

Examplex=lognrnd(0,1,1,20); ecdf(x)hold onx=lognrnd(0,1,1,10000); ecdf(x)

Extreme value distributionsNo matter what distribution you sample from, the mean of the sample tends to be normally distributed as sample size increases (what mean and standard deviation?)Similarly, distributions of the minimum (or maximum) of samples belong to other distributions.Even though there are infinite number of distributions, there are only three extreme value distribution.Type I (Gumbel) derived from normal.Type II (Frechet) e.g. maximum daily rainfallType III (Weibull) weakest link failure

Examplex=5-0.3*randn(10,1000); minx=min(x); hist(minx); ecdf(minx)

Gumbel distributionPDF and CDF

Mean, median, mode and variance

Weibull distributionProbability distribution

Used to describe distribution Of strength or fatigue life in brittle materials (weakest link connection)If it describes time to failure, then k1 indicates increasing rate.Useful for other phenomena like wind speed distribution.Can add 3rd parameter by replacing x by x-c.