M_{S_n^*} (t) &= \big{(} M_{X^*}(t/\sqrt{n}) \big{)}^n \\ \\ Thus, if the random variable X is log-normally distributed, then Y = ln (X) has a normal distribution. . Because the mgf of the normal distribution is defined at any real number, all moments for the lognormal distribution exist. The formula for a moment generating function in question is (e1) g ( t) = 0 f ( x) e t x d x where the pdf of the lognormal distribution is given by ( 1) below. On the Laplace transform of the Lognormal distribution by Sren Asmussen, Jens Ledet Jensen and Leonardo Rojas-Nandayapa. 3.10 Characteristic Function For some distributions (e.g., the Cauchy and lognormal distributions), the MGF does not exist. And now I discovered that all this (and more) was stated earlier by Cardinal Existence of the moment generating function and variance. The MGF of the lognormal distribution with parameter , and where is real, is expressed as M( ) = E(f 1) where the shifted process fe t is coming from SDE (22, 23). How to compute the second and third central moment of y, I tried number of times but may be was not able to correct the basis right. Copyright 2005 \], 17.4. 19.3.4. times-to-failure, not used as a parameter, and the standard deviation can be Why the Lognormal Distribution is used to Model Stock Prices. Does a beard adversely affect playing the violin or viola? The $c$ represents a positive real number such that $$\color{blue}{e^{tx}e^{-(\ln x)^2/2} \ge c\quad \forall x\ge k}$$ and so. + \frac{t^3}{n^{3/2}} \cdot \frac{E({X^*}^3)}{3!} Use MathJax to format equations. But first you need to derive the formula for the integral f (x)=exp (- (1/2)x^2) from negative to infinity to infinity. Can anyone help me out here? \], \[ Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? from which it follows that. I think I get the proof. $$\color{blue}{e^{tx}e^{-(\ln x)^2/2} \ge c\quad \forall x\ge k}$$. Now it makes perfect sense. Use MathJax to format equations. In Casella and Berger (2002) I found a proof for the moment-generating function (mfg) of a lognormal distribution not being existent (see exercise 2.36 on page 81 and the answer provided here on page 2-12). By Ani Adhikari It also arises in mathematical finance in the fundamental geometric Brownian motion model of asset price dynamics. Thats the m.g.f. If x = , then f ( x) = 0. $$g_{d=1}(t) = \sum_{k=0}^\infty m(k) t^k\tag{5a} $$, $$g_{d=k! The mgf is Two important variations on the previous results are worth mentioning. do not work in our case, as $\lim_{k\to\infty}{ m(k) /d(k)} \to \infty$, and hence the sums diverge for any $t>0$. Thanks for contributing an answer to Mathematics Stack Exchange! returned value is the square root of the variance of the natural logarithms It would be desirable to have a closed expression for mgf but up to now I have only derived some approximate formulae which are not very enlightning. Cookie Notice. Given the MGF of the normal distribution M_x (t)= e^ ( (t+0.5^2 t^2)). In fact, all that is needed is that Var(Xi) = 2 < 1. (4) (4) M X ( t) = E [ e t X]. and so. Importantly, the cumulative distribution function of lognormal sums is derived as an alternating series and convergence acceleration via the Epsilon algorithm is used to reduce, in some cases, the . If this is what you had in mind then I will edit my question accordingly. Suppose X has a standard normal distribution. + \cdots \Big{)}^n \\ \\ Simply because a mgf exists and will be provided below. (numeric (1)) Standard deviation of the distribution on the natural scale, defined on the positive Reals. The MGF for the binomial distribution has been calculated explicitly and used to prove the well-known formulas for the mean and variance of a binomial variable. Key is when $t>0$ and $x \to\infty$ then $e^{tx}$ tends to blow up (also see here). Using the expected value for continuous random variables, the moment . The best answers are voted up and rise to the top, Not the answer you're looking for? apply to documents without the need to be rewritten? In the present paper we introduce a new probability measure that we refer to as the star probability measure as an alternative approach to compute the moment-generating function of, A number of different ways are examined of representing the characteristic function (t) of the lognormal distribution, which cannot be expanded in a Taylor series based on the moments. Lets use this result to prove the CLT. Since this includes most, if not all, mechanical systems, the lognormal In 2 the use. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. of the normal distribution with mean X + Y and variance X 2 + Y 2. represents the mean of the natural logarithms of the times-to-failure, while This last fact makes it very nice to understand the distribution of sums of random variables. What are some tips to improve this product photo? f(x) = {e x, x > 0; > 0 0, Otherwise. The formula for the probability density function of the general Weibull distribution is. In practice, it is easier in many cases to calculate moments directly than to use the mgf. rev2022.11.7.43014. So the mean is given by yeah, this formula which is B plus A, over to where B is 99 A is zero, And this gives us a mean of 49.5. Why doesn't this unzip all my files in a given directory? So much for the mgf not existing! So X \geq 0 with probability one. \begin{align*} Some brief comments are made on the set of distributions having the same moments as a lognormal. Proof. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. What exactly are moments? Is there an industry-specific reason that many characters in martial arts anime announce the name of their attacks? ReliaSoft Corporation, ALL RIGHTS model the lives of units whose failure modes are of a fatigue-stress nature. f(x) = {1 e x , x > 0; > 0 0, Otherwise. Where are the imaginary components in a moment generating function (MGF) of a distribution? What is the distribution of \(t\)? Notice that this series is divergent and should be treated as a formal series which has to be cut off at some convenient point $k$. A linearizing transform is used with a linear minimax approximation to determine an optimal lognormal approximation to a lognorian sum distribution, which is several orders of magnitude more accurate than previous approximations. \], \[ Lognormal Distribution. The lognormal distribution is a continuous distributionon \((0, \infty)\) and is used to model random quantities when the distribution is believed to be skewed, such as certain income and lifetime variables. Since the distribution of lognormal sums is not log-normal and, In this paper it is established that the lognormal distribution is not determined by its moments. \], \[ Then the variable has a log-normal distribution with parameters and . Removing repeating rows and columns from 2d array. Proof: The probability density function of the normal distribution is f X(x) = 1 2 exp[1 2( x )2] (3) (3) f X ( x) = 1 2 exp [ 1 2 ( x ) 2] and the moment-generating function is defined as M X(t) = E[etX]. We now introduce a function which exists for any probability distribution. Can we integrate out such a term if if we evaluate it accordingly? The lognormal doesn't have an MGF; the integral needs to converge for $t$ in a neighborhood of 0, but the integral for $E(e^{tX})$ is not defined on the positive side. s d = v a r 2. . Then the mgf of \(Z\) is given by. For fixed , show that the lognormal distribution with parameters and is a scale family with scale parameter e. The result says that it is enough to show that the mgfs of the \(Y_n\)s converge to the mgf of \(Y\). Examples and counterexamples. 4.2 Discrete Probability Distributions We start with discrete probability distributions. The case where = 0 is called the 2-parameter Weibull distribution. Make sure you can explain/show why such a $c$ exists! MathJax reference. \to e^{t^2/2} \text{as } n \to \infty by ignoring small terms and using the fact that for any standardized random variable \(X^*\) we have \(E(X^*) = 0\) and \(E({X^*}^2) = 1\). Hence the formula for $t\gt0$ does NOT define a mgf. MIT, Apache, GNU, etc.) A lognormal distribution is a result of the variable " x" being a product of several variables that are identically distributed. Since a normal \((\mu, \sigma^2)\) variable can be written as \(\sigma Z + \mu\) where \(Z\) is standard normal, its m.g.f. Since the distributionof lognormal sums Is it enough to verify the hash to ensure file is virus free? $\chi^2$ function problem - moment generating functions, QGIS - approach for automatically rotating layout window, Space - falling faster than light? Namely, that the "proof" of non existence of a moment generating function of the lognormal distribution is wrong. The main example of convergence that we have seen is the Central Limit Theorem. Now the general definition of a moment generating function (mgf) can be found in (https://en.wikipedia.org/wiki/Moment-generating_function) and reads, "The moment-generating function of a random variable $X$ is the expectation of the random variable $e^{t X}$, where $t\in \mathbb {R}$, and wherever this expectation exists ". distributions. #1 Let x ~ N (,^2 ) and consider the lognormal r.v. Proof: In the Special Distribution Simulator, select the normal distribution and keep the default settings. The lognormal distribution is always bounded from below by 0 as it helps in modeling the asset prices, which are unexpected to carry negative values. Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$ denote the continuous uniform distribution on the interval $\closedint a b$.. Then the moment . Proof: Again from the definition, we can write X = e Y where Y has the normal distribution with mean and standard deviation . The quotes are because we will use the above result without proof, and also because the argument below involves some hand-waving about approximations. The logarithmic distribution (sometimes known as the Logarithmic Series distribution) is a discrete, positive distribution, peaking at x = 1, with one parameter and a long right tail. of the natural logarithms of the data. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. &= ~ e^{t^2/2} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}(z- t)^2} dz \\ \\ It only takes a minute to sign up. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. (4) (4) M X ( t) = E [ e t X]. The lognormal distribution is a basic model for describing size phenomena in economics and related fields (see, e.g., [ 12 ]), including distributions of personal income, actuarial losses, or city sizes. Removing repeating rows and columns from 2d array, Position where neither player can force an *exact* outcome. Did Great Valley Products demonstrate full motion video on an Amiga streaming from a SCSI hard disk in 1990? Third Party Privacy Notice | The answer is that the integral must be convergent, and convergence of the integral requires $t\le0$. Proposition Let be a normal random variable with mean and variance . To describe the probability distribution of a random vari-able, we introduce the CDF. The case where = 0 and = 1 is called the standard Weibull distribution. Connect and share knowledge within a single location that is structured and easy to search. For a proof, see Theorem V.7.1 on page 133 of Gut [8]. The mean value of the The formula for a moment generating function in question is, $$g(t) = \int_0^\infty f(x) e^{t x}\,dx\tag{e1}$$. Instead well just point out that it should seem reasonable. How can you prove that a certain file was downloaded from a certain website? $$g_{d=k!^k}(t) = \sum_{k=0}^\infty \frac{m(k)}{k!^k} t^k \tag{5c}$$. The newly defined distribution satisfies the following properties. where the pdf of the lognormal distribution is given by $(1)$ below. Now we are asked to find a mean and variance of X. Series B. More generally, we can compute all of the moments. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. (ii) The mean of -gamma distribution is equal to a parameter . The motivation of this study is to investigate new methods for the calculation of the moment-generating function of the lognormal distribution. $$f(x) = \frac{1}{x \sqrt(2 \pi) } \exp \left(-\frac{1}{2} \log ^2(x)\right)\tag{1}$$, A moment generating function $g(t)$ would be a sum like, $$g_d(t) = \sum_{k=0}^\infty m(k) \frac{t^k}{d(k)} \tag{4}$$. As may be surmised by the name, the lognormal distribution has certain These are not the same as mean and standard deviation, which is the subject of another post, yet they do describe the distribution, including the reliability function. apply to documents without the need to be rewritten? The claim is then that the "mgf only exists when that expectation exists for $t$ in some open interval around zero. For example, If a random variable X is considered as the log-normally distributed then Y = In(X) will have a normal distribution. From that, we can find the momment generating function as follows: $E(Y^n)$ = $\int_0^{\infty}\frac{x^n\phi(\frac{logx-\mu}{\sigma})}{\sigma x}dx$. Since the lognormal distribution is bound by zero on the lower side, it is perfect for modeling asset prices that cannot . Using the laws of exponents you then get the integral over the entire x,y-plane of f (r) where r^2=x^2+y^2. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. To learn more, see our tips on writing great answers. rev2022.11.7.43014. Proof The probability density function g of the standard logistic distribution is given by g(z) = ez (1 + ez)2, z R g is symmetric about x = 0. g increases and then decreases with the mode x = 0. g is concave upward, then downward, then upward again with inflection points at x = ln(2 + 3) = 1.317. \], \[ That is, there is a one-to-one correspondence between the r.v.'s and the mgf's if they exist. Thanks for contributing an answer to Cross Validated! In the present paper we introduce a new probability measure that we refer to as . Is there any alternative way to eliminate CO2 buildup than by breathing or even an alternative to cellular respiration that don't produce CO2? . @ DomB (1) No, the "key" is not $x\to\infty$ but $ t\le0$. \end{split}\], \[ . there are other distributions with the same sequence of moments). Starting point is the following lognormal pdf (with $\mu = 0$ and $\sigma^2 = 1$): $f(x) = \int_0^\infty \frac 1{2\pi x}e^{-(ln(x))^2/2}dx$, The mgf of the lognormal distribution is therefore, $M_x(t) = \int_0^\infty \frac {e^{tx}}{2\pi x}e^{-(ln(x))^2/2}dx$, In reference to l'Hopital rule, they then point out that, $\lim \limits_{x \to \infty} e^{tx-(ln(x))^2} = \infty.$, They conclude the proof by stating that for any $k > 0$ there is a constant $c$ such that, $\int_k^\infty \frac {e^{tx}}{x}e^{-(ln(x))^2/2}dx\ge c\int_k^\infty \frac 1xdx = c\ln|_k^\infty = \infty.$. Sums of Independent Normal Variables, 19.3.4. What is this political cartoon by Bob Moran titled "Amnesty" about? Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros. The lognormal distribution is skewed positively with a large number of small values. The result requires a careful statement and the proof requires considerable attention to detail. \end{split}\], \[ Note that the requirement of a MGF is not needed for the theorem to hold. The mean m and variance v of a lognormal random variable are functions of the lognormal distribution parameters and : m = exp ( + 2 / 2) v = exp ( 2 + 2) ( exp ( 2) 1) Also, you can compute the lognormal distribution . The mgf, if it exists, determines a distribution. Fortunately that question is, Moment Generating Function for Lognormal Random Variable, stats.stackexchange.com/questions/116644/, On the Laplace transform of the Lognormal distribution, Laplace Transforms of Probability Distributions and Their Inversions Are Easy on Logarithmic Scales, Accurate Computation of the MGF of the Lognormal Distribution and its Application to Sum of Lognormals, Uniform Saddlepoint Approximations and Log-Concave Densities, Existence of the moment generating function and variance, Mobile app infrastructure being decommissioned. The pdf starts at zero, increases to its mode, and decreases thereafter. Mean of a function of $K$ Gaussian random variables, Moment Generating Function of a nonlinear transformation of an exponential random variable, How to derive the cdf of a lognormal distribution from its pdf, Plot the density function of a normal random variable knowing only the characteristic function in R. Different $\mathbb{E}[x]$ depending on choice of substitution? The probability-density function of the sum of lognormally distributed random variables is studied by a method that involves the calculation of the Fourier transform of the characteristic function; Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. Since. \begin{align*} \], \[\begin{split} The Maxwell distribution, named for James Clerk Maxwell, is the distribution of the magnitude of a three-dimensional random vector whose coordinates are independent, identically distributed, mean 0 normal variables. normal.mgf <13.1> Example. What does the integral of a function times a function of a random variable represent, conceptually? lognormal distribution is not uniquely determined by its moments as seen in [8] for some multiplicity . Hence 1 / X = e Y . However, after that, I'm a bit lost towards exactly what to do. of the normal distribution with mean \(\mu_X + \mu_Y\) and variance \(\sigma_X^2 + \sigma_Y^2\). That's the m.g.f. For every \(n \ge 1\) let \(S_n = X_1 + X_2 + \cdots + X_n\). Sums of lognormal random variables (RVs) are of wide interest in wireless communications and other areas of science and engineering. A tag already exists with the provided branch name. It must be related to $e^{tx}e^{-(ln(x))^2/2}$. Let us refer to this distribution as xB( + ; ). Furthermore, X 1 and X 2 are uncorrelated if and only if they are independent. The random variables \(X_i^*\) are i.i.d., so let \(M_{X^*}\) denote the mgf of any one of them. Conditional Expectation As a Projection, 24.3. The lognormal distribution has two parameters, , and . To learn more, see our tips on writing great answers. because the integral is \(1\). and standard deviation, the user is reminded that these are given as the Let \(X_1, X_2, \ldots\) be i.i.d. That is, the parameter There is no exact formula for the mgf, but that paper gives good approximations. Specifically, the which gives us the estimates for and based on the method of moments. ( a) Lognormal distribution: X is lognormal if X = e^Y for some normal random variable Y. The exponential in M X(t) expands to . represents the distribution can have widespread application. What I do not get is what constant $c$ is supposed to represent. Stack Overflow for Teams is moving to its own domain! \begin{align*} For the evaluation of the moments of the generalized Lognormal distribution, the following holds. The parametric functions m 1 and v The lognormal distribution is commonly used to The case $t\gt0$ was already studied in the answers of others and was shown to violate the condiditon that the expectation exists (because the integral diverges). M_Z(t) ~ = ~ e^{t^2/2} ~~~ \text{for all } t The moment generating function (mgf) of (or ), denoted by , is provided this expectation exists for in some neighborhood of 0. Will it have a bad influence on getting a student visa? M_X(t) = \E e^{tX}. &= ~ e^{t^2/2} Normal approximation to the Lognormal Distribution; Normal approximations to other . Lognormal Distribution Parameters in The distribution has a number of applications in settings . (clarification of a documentary), Concealing One's Identity from the Public When Purchasing a Home. It is common in statistics that data be normally distributed for statistical testing. I understand the following: Our CDF is $\Phi(\frac{logx - \mu}{\sigma})$, and thus our PDF is $\phi(\frac{logx-\mu}{\sigma})$/$\sigma y$. Since mgfs determine distributions, its not difficult to accept that if two mgfs are close to each other then the corresponding distributions should also be close to each other. Can FOSS software licenses (e.g. I understand what the final answer is and how I should get there, but I don't know exactly how to complete the square/substitute in this integral. A continuous random variable X is said to have an exponential distribution with parameter if its probability denisity function is given by. Connect and share knowledge within a single location that is structured and easy to search. Let \(Z\) be standard normal. Again, why do you assume the (unfortunate) case that $t\gt0$? M_X(t) = \E e^{tX}. [1] The moment generating function (mgf), as its name suggests, can be used to generate moments. For instance, from what Cardinal proves there, one can conclude that the lognormal do not have exponentially decaying tails. where for each \(i\), the random variable \(X_i^*\) is \(X_i\) in standard units. UW-Madison (Statistics) Stat . The Journal of the Australian Mathematical Society. Because of this inequality, since $k > 0$, we get the integral inequality obtained in your post. very similar for these two distributions. m(2)++ \frac{(-s)^k}{k!} Light bulb as limit, to what is current limited to? S_n^* = \frac{S_n - n\mu}{\sqrt{n}\sigma} = \sum_{i=1}^n \frac{1}{\sqrt{n}} \big{(} \frac{X_i - \mu}{\sigma} \big{)} = \sum_{i=1}^n \frac{1}{\sqrt{n}} X_i^* We emphasize that it is important to understand the meanings and roles that parameters play in each parametric distribution. Theorem 3.15 . HBM Prenscia.Copyright 1992 - document.write(new Date().getFullYear()) HOTTINGER BRUEL & KJAER INC. Hence the integral exists and therefore it is a valid definition of the mgf. The best answers are voted up and rise to the top, Not the answer you're looking for? \], \[\begin{split} (clarification of a documentary). Theorem 2.3.11 Let X and Y be random variables with cdfs FX and FY, respectively. t^k\tag{5b} $$. The general formula for the probability density function of the lognormal distribution is where is the shape parameter (and is the standard deviation of the log of the distribution), is the location parameter and m is the scale parameter (and is also the median of the distribution). I will come back here with a more complete answer, but for the moment I will just point to some papers. Its moment generating function equals exp(t2=2), for all real t, because Z 1 1 ext e x2= 2 p 2 dx= 1 p 2 Z 1 1 exp (x t)2 2 + t 2 dx = exp t2 2 : For the last equality, compare with the fact that the N(t;1) density inte-grates to 1. Weibull++. And consequently it can't be used to "prove" that the lognormal distribution has no mgf. What is this political cartoon by Bob Moran titled "Amnesty" about? Suppose M(t) is the moment generating function of the distribution of X. Can an adult sue someone who violated them as a child? The Central Limit Theorem says that for large \(n\), the distribution of the standardized sum. This video shows how to derive the Mean, the Variance & the Moments of Log-Normal Distribution in English.Please don't forget to like if you like it and subs. \], \[ How do you prove lognormal distribution? Details aside, what this formula is saying is that if a moment generating function is \(\exp(c_1t + c_2t^2)\) for any constant \(c_1\) and any positive constant \(c_2\), then it is the moment generating function of a normally distributed random variable. So any help will really appreciated. About HBM Prenscia | By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. How are they derived? Can FOSS software licenses (e.g. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. A wide variety of methods have been employed to, IEEE Transactions on Vehicular Technology. Consequently, by recognizing the form of the mgf of a r.v X, one can identify the distribution of this r.v. QGIS - approach for automatically rotating layout window, Find a completion of the following spaces. And for the lognormal this only exists for $t\le 0$. Copyright 2022. For example, the mathematical reasoning for the construction Theorem 2.1. 2)0 has a bivariate normal distribution so that the components of X, namely X 1 and X 2, are each normally distributed. RESERVED, The weibull.com reliability engineering resource website is a service of The limit is the moment generating function of the standard normal distribution. 14. I'd like to point out more clearly what I addressed in my comments. Does subclassing int to forbid negative integers break Liskov Substitution Principle? If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist. Note that the mean and variance of xunder B( + ; ) are and 2 respectively. It only takes a minute to sign up. Moments of the -Order Lognormal Distribution. "Proof" of the Central Limit Theorem Another important reason for studying mgf's is that they can help us identify the limit of a sequence of distributions. Interestingly, the lognormal is an example of a distribution with a finite moment sequence that is not characterized by that set of moments (i.e. Conditioning and the Multivariate Normal, 19.3.3. Cheers, $\lim_{k\to\infty}{ m(k) /d(k)} \to \infty$, Proof that the moment generating function of a lognormal distribution does not exist, en.wikipedia.org/wiki/Moment-generating_function, https://en.wikipedia.org/wiki/Moment-generating_function, Mobile app infrastructure being decommissioned, Proving that the lognormal distribution has no moment generating function, Show that the moment generating function does not exist, Integration by Substitution in $\int_0^{\infty}x^r\frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}[\sin(2\pi\log x)]dx$, Moment Generating Function of beta ( Hard ), Determining a random variable through the Taylor expansion of its moment generating function, Computing Joint Moment Generating Function, Moment generating function for a standardized sum of random variables. understood, but then how would I find E(Y^n)? . The lognormal distribution is a distribution skewed to the right. &= ~ e^{t^2/2} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}(z^2 - 2tz + t^2)} dz \\ \\ It is the normal \((t, 1)\) density integrated over the whole real line. Accurate Computation of the MGF of the Lognormal Distribution and its Application to Sum of Lognormals C. Tellambura, Senior Member, IEEE, and D. Senaratne AbstractSums of lognormal random variables (RVs) are of wide interest in wireless communications and other areas of science and engineering. Can a black pudding corrode a leather tunic? You plug in $k$ and that way it becomes the lower bound. 13. University of Alabama in Huntsville via Random Services. (It's related to the fact that for any $t>0$, we have $e^{tx}e^{-(\ln x)^2/2} \to \infty$ as $x\to \infty$.) We will give a proof of this result in Chapter 4 for the multivariate case, after we introduce the characteristic functions. In the definition of the mgf (. If the MGF existed in a neighborhood of 0 this could not occur. Here the sum is convergent for any $t$ and the moments can be found from the $k$-th derivative at $t=0$. Here's a graph of the mgf obtained by numerically integrating over $x$, The moments (calculated in my original post) are retrieved by the expansion, $$g(s) = 1+\frac{(-s)}{1! Proof of (i). Is it because we can plug $k$ into $e^{tx}e^{-(ln(x))^2/2}$ and that way it is not dependent on $x$ anymore? I'm working through the proof of a lognormal random variable and am having some difficulty in moving through it. Here is another nice feature of moment generating functions: Fact 3. R(t) = 1 ( ln(t) ) R ( t) = 1 ( ln ( t) ) The lognormal distribution is also known as a logarithmic normal distribution. If the mgf exists (i.e., if it is finite), there is only one unique distribution with this mgf.
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