Normal distribution Lognormal distribution Mean median and mode Tails Extreme value distributions Normal Gaussian distribution P robability density function PDF What does figure tell about the cumulative distribution function ID: 536903
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Slide1
Probability distribution functions
Normal distributionLognormal distributionMean, median and modeTailsExtreme value distributionsSlide2
Normal (Gaussian) distribution
Probability density function (PDF)
What does figure tell about the cumulative distribution function
(CDF)?Slide3
More on the normal distribution
Normal distribution is denoted
, with the square giving the variance.
If X is
normal,
Y=aX+b is also normal.
What would be the mean and standard deviation of Y?Similarly, if X and Y are normal variables, any linear combination, aX+bY
is also normal.Can often use any function of a normal random variables by using a linear Taylor expansion.Example: X=N(10,0.52) and Y=X2
. Then
Y
N(100,10
2)
Slide4
Estimating mean and standard deviation
Given a sample from a normally distributed variable, the sample mean is the best linear unbiased estimator (BLUE) 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
For example, for a sample of 5 from a standard normal distribution, the standard deviation will be estimated on average as 0.94 (with standard deviation of 0.34)Slide5
Lognormal distribution
If ln(X) has normal distribution X has lognormal distribution. That is, if X is normally distributed
exp
(X) is
lognormally distributed.Notation:
PDF
Mean and variance
Slide6
Mean, mode and median
Mode (highest point) =
Median (50% of samples)
Figure for
=0.
Slide7
Light and heavy tails
Normal distribution has light tail; 4.5 sigma is equivalent to 3.4e-6 failure or defect probability.
Lognormal can have heavy tail
Slide8
Fitting distribution to data
Usually fit CDF to minimize maximum distance (Kolmogorov-Smirnoff test)Generated 20 points from N(3,1
2
).
Normal fit N(3.48,0.932)Lognormal
lnN(1.24,0.26)Almost same mean andstandard deviation.Slide9
Extreme value distributions
No 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 rainfall
Type III (Weibull) weakest link failureSlide10
Maximum of normal samples
With normal distribution, maximum of sample is more narrowly distributed than original distribution.
Max of 10 standard normal samples. 1.54 mean, 0.59 standard deviation
Max of
100
standard normal samples.
2.50
mean,
0.43 standard deviationSlide11
Gumbel
distribution.Mean, median, mode and varianceSlide12
Weibull
distributionProbability distributionIts log has
Gumbel
dist.
Used to describe distribution
of
strength or fatigue life in brittle
materials.
If it describes time to failure, then k<1 indicates that failure rate decreases with time,
k=1 indicates constant rate, k>1 indicates increasing rate.Can
add 3
rd
parameter by replacing x by x-c.Slide13
Exercises
Find how many samples of normally distributed numbers you need in order to estimate the mean and standard deviation with an error that will be less than 10% of the true standard deviation most of the time.Both the lognormal and Weibull distributions are used to model strength. Find how closely you can approximate data generated from a standard lognormal distribution by fitting it with
Weibull
.
Take the introduction and preamble of the US Declaration of Independence, and fit the distribution of word lengths using the K-S criterion. What distribution fits best? Compare the graphs of the CDFs. Compare to a more contemporary text.