Other_PDFs - Other PDFs Author: John M. Cimbala, Penn State...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Other PDFs Author: John M. Cimbala, Penn State University Latest revision: 30 January 2008 Introduction There are other useful standard distributions and PDFs besides the Gaussian PDF. These include the binomial, chi-squared, exponential, gamma, lognormal, Poisson, student’s t , uniform, and Weibull PDFs. We discuss some of these in this learning module, although not in as much detail as for the Gaussian (normal) distribution. Lognormal PDF A lognormal PDF is defined as a PDF that becomes Gaussian when the x-axis is plotted as a log scale . o Lognormal PDFs often appear in air quality measurements, e.g., the size distribution of particles. It is also useful for some life and durability analyses of components and equipment or instruments. o When the PDF is plotted as usual (linear x scale), it is skewed towards the left (lower values), and has a very long tail to the right (higher values). This is shown on the first plot below. o However, when the PDF is plotted with a logarithmic x scale, all else being equal, it is no longer skewed, but becomes symmetric. In fact, it’s bell shape is identical to that of a Gaussian or normal PDF. This is shown on the second plot below. o Another way to plot lognormal PDFs is to first convert the x values to log 10 ( x ) or ln( x ), and then plot using a linear abscissa scale. Either way, the PDF again looks like a standard Gaussian PDF, as illustrated below. o To calculate statistics with a lognormal PDF, we substitute either log 10 ( x ) or ln( x ) as our variable instead of x itself. For example, if the data are for particle diameter D p in units of microns ( μ m), we let our statistics variable be x = ln[ D p /(1 μ m)] instead of D p itself. All statistics are then based on x as usual.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Student’s t PDF A student’s t PDF (also sometimes called simply the t PDF ) is similar to the Gaussian (normal) PDF, but is used for small sample sizes (typically when n < 30 , where n is the number of data points in the sample). o In simple terms, when n is small, the sample mean and sample standard deviation may differ from the population mean and population standard deviation by some unknown amount. So, for a specified confidence level (typically 95%), the student’s t PDF is expected to be wider than the normal (Gaussian) PDF. o Mathematically, the statistic called student’s t is defined as / x t Sn μ = , where x is the sample mean, S is the sample standard deviation, and n is the number of data points in the sample. is the population mean or expected value, as defined previously, but is not necessarily known. (This is the whole point of the student’s t analysis in the first place – we want to establish some confidence level in predicting .) o Statisticians use a parameter called degrees of freedom , with notation df (we do not use italics here so as to not confuse df with df , the differential of some variable f ). Note that some authors use f or v (lower case italic V) as their notation for degrees of freedom.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/23/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

Page1 / 8

Other_PDFs - Other PDFs Author: John M. Cimbala, Penn State...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online