The Lognormal Distribution Engr 323 Geppert page 1of 6 The Lognormal Distribution


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1 Engr 33 Geert age 1of 6 The Lognormal Distribution In general, the most imortant roerty of the lognormal rocess is that it reresents a roduct of indeendent random variables. (Class Handout on Lognormal Processes) A lognormal rocess is one in which the random variable of interest results from the roduct of many indeendent random variables multilied together Often when natural rocesses are concerned the lognormal rocess is alicable. Some eamles of common henomena that can be reresented by the lognormal distribution are: The occurrence of alien visitations (the rate varies with the changing oulation of cows on the lanet and increases by a roortion of the bovine oulation) leaf growth (the area of the leaf increases by some random roortion) yearly oulation growth (the growth rate is a random variable because the growth rate varies in resonse to annual fluctuations in economic, health and social conditions) interest on a savings account (comounded daily by a varying national interest rate and the amount increases by some roortion of the initial amount) fied amount of tracer ollutant in a ond some days later (the flow of the water through the ond varies from hour to hour; therefore the dilution factor (roortion) varies by some roortion of the initial concentration ) The concet I m attemting to convey is change by a roortion that can vary. The imortant formulas related to the lognormal are; (all equations can be found on the class handout, Lognormal Processes ) Probability Density Function (PDF) f ( ) = σ 1 ln π e ln µ ln.5 σ ln < µ < o / w Eected Value α = E( X ) = µ Variance σ ln µ ln + σ ln ( e 1) µ ln + σ ln β = V ( X ) = σ note: these show the relationshi between the normal and lognormal values.
2 Engr 33 Geert age of 6 Problem Statement:? What are the Lognormal distribution arameters? If we are given α & β, (these are the µ & σ ) we must rewrite the above equations in terms of µ ln & σ ln, which are the arameters of the Lognormal Distribution. 1 µ ln = ln µ σ ln σ = + σ ln ln 1 µ l note: the lognormal standard deviation, σ ln, must be comuted first It is given that µ = 6.67 & σ = 1.73 But we want: X~Lognormal(µ ln, σ ln ) Therefore, aly the equations: σ = σ ln ln 1 µ = ln 1 + =.4915 = σ ln l µ ln = ln µ σ ln = ln = = µ ln X~Lognormal(1.7145,.4915)??What are the 1, 1, 5, 9, 99 th ercentiles of the Lognormal? The ercentile reresents the value of at which % of the oulation is below. Mathematically, P( X ) = There are several ways to determine the value of corresonding to a given ercentile value for the lognormal distribution. Method 1: (Use the Standard Normal Tables) As shown in class: Standardization of Ln using the Ztables: ln µ ln z = σ ln Rearranging to solve in terms of : z σ ln +µ LNX **HINT** You had better have this equation on your cheat sheet!!
3 Engr 33 Geert age 3of 6 Now we find the z value corresonding to the ercentile that we are interested in and back calculate for For eamle, say we want the 37 th ercentile, we d look u.37 on the Ztable and find that the z having this robability is.33. Plugging that value and the z LNX Lognormal arameters into σ ln +µ we d get the value (.33* ).37 = 4.7 For the answers to our secific roblem refer to Table 1 1 Method  Use EXCEL Using the Ecel sreadsheet can be retty quick and easy too. What you have to is use the CDF function, (which is all Ecel has for the lognormal anywaythe PDF is not an intrinsic function. If you want to grah the PDF you must enter it in manually. To learn more about this, check out the section on the PDF grah) Stes: 1) You ll enter into a cell the CDF function: =LOGNORMDIST(,µ ln,σ ln ) For eamle: =LOGNORMDIST($A1,$B$3,$B$4), where $A1 is the column of values, $B$3, & $B$4 are the lognormal arameters,µ ln & σ ln resectively ) Then choose (under Tools ) Goal Seek. The dialog bo will have Set Cell that cell should be where you have the CDF function. Enter in the decimal value of the ercentile of interest in the To Value cell The By Changing cell should reference the cell where you want the, $A1, value to aear. Also, this cell should be the cell referenced in your CDF equation. And as if by magic, you receive your answerbeautiful!actually, it s Numerical Analysis wizardry at work (iteration after iteration)! Consult a numerical analysis tet for further details if you have a burning curiosity about the secific algorithm that Ecel uses. ***To note: You may get a resonse of no solution if you haven t entered in an initial guess or there s a crazy value in the cell try 1 in each cell and it should work fine. 1 Located under Method
4 Engr 33 Geert age 4of 6 The solutions I got were slightly different than those achieved manually. That s robably just roundoff or truncation error in one of the algorithms. Table 1 lists the values achieved by both methods. Table 1 Percentile Values calculated by Mathematical and Numerical Techniques Percentile z Manual value Ecel value 1 st z.1 = = th z.1 = = th z.5 =.5 = th z.9 = = th z.99 = = Method 3 Use Integration Integrate the PDF but whoawhy when there are other less ainful ways? Well, unfortunately, sometimes the CDF and tables are not available and we must do this. (Hey and why else did we take 3 semesters of Calculus?) F ( ) = uσ ln u µ ln.5 1 σ ln e ln π This function can be integrated mathematically by using integration by arts OR numerically by your friend the calculator. du
5 Engr 33 Geert age 5of 6 Grahs of the PDF and CDF of the Lognormal Figure 1 is the grah of the robability density function of the lognormal. LogNormal Probability Density Function E(X) = 6.67 V(X) = f() = Value of Random Variable X Figure 1 PDF of the Lognormal Distribution Unlike the symmetry observed in the normal distribution, the lognormal is characterized by its skewidness (that means it s skewed), a single mode and a long tail to the right. The Lognormal Cumulative Density Function is illustrated in Figure. Notice that the 5 th ercentile is NOT E(X) = Rather it is This is so because of the skewidness of the lognormal. (imortant concet) Logormal Cummulative Density Function E(X) = 6.67 V(X) = 1.73 F() Figure  Cumulative Density FunctionLognormal Distribution. The arrows (left to right) indicate the 1 st, 1 th, 5 th, 9 th, and 99 th ercentiles. Refer to Table 1 for numerical values Some words may have been created by Geie
6 Engr 33 Geert age 6of 6 Lastly, Figure 3 illustrates the relative shae and location of the df s of the lognormal and the normal distributions. PDF of the Normal and Lognormal Distributions E(X) = 6.67 V(X) = 1.73 f() = the robability associated with = value of the Random Variable X normaldf lognormaldf
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