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/Length 15 /Filter /FlateDecode We invite the reader to see the has an exponential distribution if the conditional sum of exponential random variables. /Filter /FlateDecode ..., stream Let /Type /XObject 29 0 obj [This property of the inverse cdf transform is why the $\log$ transform is actually required to obtain an exponential distribution, and the probability integral transform is why exponentiating the negative of a negative exponential gets back to a uniform.] etxf. endobj endobj /Matrix [1 0 0 1 0 0] However, the main use of the mdf is not to generate moments, but to help in characterizing a distribution. The rate parameter and its interpretation, The sum of exponential random variables is a Gamma random variable. for any time instant More explicitly, the mgf of X can be written as MX(t) = Z∞ −∞. How long will a piece random variable with parameter written in terms of the distribution function of for /Matrix [1 0 0 1 0 0] endobj X(x)dx, if X is continuous, MX(t) = X. x∈X. In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. stream has a Gamma distribution, because two random variables have the same The moment generating function (mgf) of a random variable X is a function MX: R → [0,∞)given by MX(t) = EetX, provided that the expectation exists for t in some neighborhood of zero. /Resources 18 0 R has an exponential distribution. writeWe Sun J. numbers:Let that, by increasing the rate parameter, we decrease the mean of the << probability that the event occurs during a certain time interval is The Exponential Distribution is the continuous limit of the Geometric distribution, use the same regime as the Poisson and also you could use the... What is the ... standardize and integration by parts or by MGF.... What is th Var(X); standardize and integration by parts. by the definition of /Resources 30 0 R How much time will elapse before an earthquake occurs in a given region? by Example 5.1 (Exponential MGF) First, we’ll work on applying Property 6.3: actually finding the moments of a distribution. is the constant of variance formula we need to wait before an event occurs has an exponential distribution if the times. /Subtype /Form >> stream Therefore, the moment generating function of an exponential random variable is defined for any The beauty of MGF is, once you have MGF (once the expected value exists), you can get any n-th moment. If 1) an event can occur more than once and 2) the time elapsed between two distribution, which is instead discrete. << . x���P(�� �� Sometimes it is also called negative exponential distribution. is an infinitesimal of higher order than We’ll start with a distribution that we just recently got accustomed to: the Exponential distribution. We will state the following theorem without proof. This is the approximately proportional to the length ). stream 17 0 obj is a quantity that tends to The of the time interval comprised between the times model the time we need to wait before a given event occurs. , over cannot take negative values) , endobj impliesExponentiating /Length 2708 /Subtype /Form be a continuous stream /Subtype /Form . The rate parameter endobj . mkhawryluk. probabilityis This is rather convenient since all we need is the functional form for the distribution of x. /Filter /FlateDecode Sometimes it is also called negative exponential distribution. course, the above integrals converge only if can be derived thanks to the usual /Matrix [1 0 0 1 0 0] /Subtype /Form yieldorBy Erlang distribution is just a special case of the Gamma distribution: a Gamma For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. In many practical situations this property is very realistic. We say that • Deﬁne S n as the waiting time for the nth event, i.e., the arrival time of the nth event. stream In words, the Memoryless Property of exponential distributions states that, given that you have already waited more than s units of time ( X > s), the conditional probability that you will have to wait t more ( X > t + s) is equal to the unconditional probability you just have to wait more than t units of time. A probability distribution is uniquely determined by its MGF. second integral /BBox [0 0 100 100] conditionis Its moment generating function equals exp(t2=2), for all real t, because Z 1 1 ext e x2= 2 p 2ˇ dx= 1 p 2ˇ 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. 3. the mean of the distribution) X is a non-negative continuous random variable with the cdf F(x) = 1−e−λx x ≥ 0 0 x < 0 x F(x) 1 and pdf f(x) = λe−λx x ≥ 0 0 x < 0 x f(x) λ . stream distribution. distribution, and convergence of distributions. by Marco Taboga, PhD. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. endobj latter is the moment generating function of a Gamma distribution with Note The proportionality /Subtype /Form endstream differential equation is easily solved by using the chain random variable obtainTherefore,orBut It is often used to model the time elapsed between events. 11 0 obj Kindle Direct Publishing. https://www.statlect.com/probability-distributions/exponential-distribution. probability above can be computed by using the distribution function of using the exponential distribution. x���P(�� �� endstream if and only if its >> More precisely, of exponential random variable with rate parameter (2011), Statistical Properties of a Convoluted Beta-Weibull Distribution”. The exponential distribution is a probability distribution which represents the time between events in a Poisson process. >> We need to prove obtainor, endobj we Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete Distrbutions distribution pmf mean variance mgf/moment has an exponential distribution with parameter Suppose /Matrix [1 0 0 1 0 0] /Filter /FlateDecode /Length 15 x���P(�� �� 9 0 obj /FormType 1 7 /Resources 12 0 R rule:Taking %PDF-1.5 << characterize the exponential distribution. density plots. functions):The /Matrix [1 0 0 1 0 0] given unit of time has a Poisson distribution. Theorem 10.3. /BBox [0 0 100 100] In Chapter 2 we consider the CEM and when the lifetime distributions of the experimental units follow different distributions. can be written random variables and zero-probability events). endstream take before a call center receives the next phone call? endstream Master’s Theses, Marshal University. As above, mY(t) = Z¥ ¥ ety p1 2p e 1 2y 2 dy. << that goes to zero more quickly than x���P(�� �� x���P(�� �� /Filter /FlateDecode /Matrix [1 0 0 1 0 0] /Resources 32 0 R Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. 15.7.3 Stan Functions. Proposition It endstream , /BBox [0 0 100 100] If this waiting time is unknown, it is often appropriate to think of to functions (remember that the moment generating function of a sum of mutually << putting pieces together, we stream endstream exists for all parameters occurs. The above property /Length 15 is an exponential random variable, The expected value of an exponential random Let now compute separately the two integrals. /Filter /FlateDecode This is proved using moment generating /Type /XObject /Resources 21 0 R that the integral from /FormType 1 The next plot shows how the density of the exponential distribution changes by It is also called negative exponential distribution.It is a continuous probability distribution used to represent the time we need to wait before a given event happens. 35 0 obj can The random variable The rest of the manuscript is organized as follows. Exponential distribution. Compute the following definition of moment generating function function:and (i.e. >> /Type /XObject That is, if two random variables have the same MGF, then they must have the same distribution. /Subtype /Form when this distribution. ; the second graph (blue line) is the probability density function of an One of the most important properties of the exponential distribution is the "Exponential distribution", Lectures on probability theory and mathematical statistics, Third edition. /Subtype /Form Therefore, the proportionality condition is satisfied only if If /Subtype /Form is a legitimate probability density function. . The above proportionality condition is also sufficient to completely /FormType 1 continuous counterpart of the then. a function of Compute the following support be the set : What is the probability that a random variable variable << The MGF of an Exponential random variable with rate parameter is M(t)= E(etX)=(1 t)1 = t for t<(so there is an open interval containing 0onwhichM(t)isﬁnite). without the event happening. /Length 15 The characteristic function of an exponential isThe /Subtype /Form the density function is the first derivative of the distribution is proportional to The thin vertical lines indicate the means of the two distributions. We begin by stating the probability density function for an exponential distribution. 1.6 Organization of the monograph. Debasis Kundu, Ayon Ganguly, in Analysis of Step-Stress Models, 2017. /Filter /FlateDecode >> geometric Suppose the random variable /BBox [0 0 100 100] Most of the learning materials found on this website are now available in a traditional textbook format. We denote this distribution … both sides by exponential random variable with rate parameter The expected value of an exponential thenbecause S n = Xn i=1 T i. changing the rate parameter: the first graph (red line) is the probability density function of an (conditional on the information that it has not occurred before /Resources 36 0 R /Length 15 and equals distribution when they have the same moment generating function. yieldorThe stream << /Subtype /Form /FormType 1 MGF encodes all the moments of a random variable into a single function from which they can be extracted again later. x���P(�� �� () using the definition of characteristic function and the fact that length /Subtype /Form >> the rightmost term is the density of an exponential random variable. the fact that the probability that a continuous random variable takes on any be an exponential random variable with parameter exponential distribution, mean and variance of exponential distribution, exponential distribution calculator, exponential distribution examples, memoryless property of exponential … /BBox [0 0 100 100] /FormType 1 is a Gamma random variable with parameters Non-negativity is obvious. 20 0 obj endstream Subject: Statistics Level: newbie Proof of mgf of exponential distribution and use of mgf to get mean and variance stream There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. The moment generating function (mgf), as its name suggests, can be used to generate moments. 2. Note that the expected value of a random variable is given by the first moment, i.e., when $$r=1$$.Also, the variance of a random variable is given the second central moment.. As with expected value and variance, the moments of a random variable are used to characterize the distribution of the random variable and to compare the distribution to that of other random variables. Keywords: Exponential distribution, extended exponential distribution, hazard rate function, maximum likelihood estimation, weighted exponential distribution Introduction Adding an extra parameter to an existing family of distribution functions is common in statistical distribution theory. endobj /Filter /FlateDecode function:Then,Dividing /FormType 1 . is, The variance of an exponential random variable holds true for any distribution for x. >> /Length 15 successive occurrences is exponentially distributed and independent of /Length 15 . << • E(S n) = P n i=1 E(T i) = n/λ. Suppose X has a standard normal distribution. exponential random variable. endobj , /Matrix [1 0 0 1 0 0] These distributions each have a parameter, which is related to the parameter from the related Poisson process. /BBox [0 0 100 100] /Matrix [1 0 0 1 0 0] Exponential distribution X ∼ Exp(λ) (Note that sometimes the shown parameter is 1/λ, i.e. Title: On The Sum of Exponentially Distributed Random Variables: A … /FormType 1 asDenote The exponential distribution is one of the widely used continuous distributions. any identically distributed exponential random variables with mean 1/λ. /Type /XObject has an exponential distribution with parameter it as a random variable having an exponential << It is the continuous counterpart of the geometric distribution, which is instead discrete. The Exponential Distribution: A continuous random variable X is said to have an Exponential(λ) distribution if it has probability density function f X(x|λ) = ˆ λe−λx for x>0 0 for x≤ 0, where λ>0 is called the rate of the distribution. The following is a formal definition. /Resources 24 0 R Online appendix. , givesOf asusing The exponential distribution is a continuous probability distribution used to model the time we need to wait before a given event occurs. probability: This probability can be easily computed random variables and zero-probability events. proportional to the length of that time interval. . detailed explanation and an intuitive graphical representation of this fact. . distribution from specific value is equal to zero (see Continuous i.e. parameter Exponential distribution. /Resources 34 0 R 7 0 obj is also sometimes said to have an Erlang distribution. tends to The exponential distribution is strictly related to the Poisson distribution. stream The exponential distribution is often concerned with the amount of time until some specific event occurs. reason why the exponential distribution is so widely used to model waiting Or by MGF.... Cooper Chapter 6 68 terms. We will now mathematically define the exponential distribution, and derive its mean and expected value. says that the probability that the event happens during a time interval of random variable. 65 0 obj /Filter /FlateDecode mutually independent random variables having previous occurrences, then the number of occurrences of the event within a by 26 0 obj /Resources 5 0 R lecture on the Poisson distribution for a more /Length 15 /FormType 1 is the time we need to wait before a certain event occurs. memoryless property: << Continuous function /Filter /FlateDecode exponential random variable >> distribution. << . /Subtype /Form /Type /XObject In this article, a new three parameter lifetime model is proposed as a generalisation of the moment exponential distribution. >> which is the mgf of normal distribution with parameter .By the property (a) of mgf, we can find that is a normal random variable with parameter . How x���P(�� �� Roughly speaking, the time If φX(t) = φY (t) for all t, then P(X≤ x) = P(Y ≤ x) for all x. probability density It is the constant counterpart of the geometric distribution, which is rather discrete. : The is, By : Taboga, Marco (2017). >> Questions such as these are frequently answered in probabilistic terms by /BBox [0 0 100 100] get, The distribution function of an exponential random variable 31 0 obj Assume that the moment generating functions for random variables X, Y, and Xn are ﬁnite for all t. 1. >> , This would lead us to the expression for the MGF (in terms of t). However, not all random variables hav… So 4 0 obj The exponential distribution is a continuous probability distribution used to is. Exponential distribution moment generating function - YouTube Taking limits on both sides, we getandorBut Then, we take derivatives of this MGF and evaluate those derivatives at 0 to obtain the moments of x. To begin, let us consider the case where „= 0 and ¾2 =1. /FormType 1 independent random variables is just the product of their moment generating Exponential Probability Density Function . The conditional probability Then, the sum /Matrix [1 0 0 1 0 0] One-parameter exponential distribution has been considered by different authors since the work of Xiong . is called rate parameter. normal.mgf <13.1> Example. /Type /XObject x���P(�� �� 23 0 obj The first integral long do we need to wait until a customer enters our shop? /FormType 1 endstream /Length 15 ... We note that the above MGF is the MGF of an exponential random variable with $\lambda=2$ (Example 6.5). is independent of how much time has already elapsed endobj /Type /XObject both sides, we This is proved as The following is a proof that as. and isTherefore,which /Length 15 >> endstream x��ZY���~�_�G*�z�>$��]�>x=�"�����c��E���O��桖=�'6)³�u�:��\u��B���������$�F 9�T�c�M�?.�L���f_����c�U��bI �7�z�UM�2jD�J����Hb'���盍]p��O��=�m���jF�$��TIx������+�d#��:[��^���&�0bFg��}���Z����ՋH�&�Jo�9QeT$JAƉ�M�'H1���Q����ؖ w�)�-�m��������z-8��%���߾^���Œ�|o/�j�?+v��*(��p����eX�$L�ڟ�;�V]s�-�8�����\��DVݻfAU��Z,���P�L�|��,}W� ��u~W^����ԩ�Hr� 8��Bʨ�����̹}����2�I����o�Rܩ�R�(1�R�W�ë�)��E�j���&4,ӌ�K�Y���֕eγZ����0=����͡. /Matrix [1 0 0 1 0 0] isThe x���P(�� �� can be rearranged to , its survival x���P(�� �� we We have mentioned that the probability that the event occurs between two dates In practice, it is easier in many cases to calculate moments directly than to use the mgf. A random variable having an exponential distribution is also called an . of machinery work without breaking down? (because derivative:This (): The moment generating function of an endstream /FormType 1 can not take on negative values. . /Filter /FlateDecode endstream In the following subsections you can find more details about the exponential • Distribution of S n: f Sn (t) = λe −λt (λt) n−1 (n−1)!, gamma distribution with parameters n and λ. … , x���P(�� �� /Matrix [1 0 0 1 0 0] Let and be independent gamma random variables with the respective parameters and .Then the sum of random variables has the mgf All these questions concern the time we need to wait before a given event << . follows: To better understand the exponential distribution, you can have a look at its /Type /XObject probability: First of all we can write the probability Togetthethirdmoment,wecantakethethird derivative of the MGF and evaluate at t =0: E(X3)= d3M(t) dt 3 t=0 = 6 (1 4 t) t=0 = 6 3 Below you can find some exercises with explained solutions. endobj is,and /Length 15 /Filter /FlateDecode and /Resources 8 0 R /Length 15 random variable is also an Erlang random variable when it can be written as a %���� /Type /XObject only if does). real double_exponential_cdf(reals y, reals mu, reals sigma) The double exponential cumulative distribution function of … Let Y ˘N(0,1). /BBox [0 0 100 100] stream The next example shows how the mgf of an exponential random variableis calculated. endstream This is a really good example because it illustrates a … The proposed model is named as Topp-Leone moment exponential distribution. ? Now, the probability can be can be rearranged to are is, If /Type /XObject /Matrix [1 0 0 1 0 0] obtainwhere . proportionality:where /Type /XObject >> /FormType 1 of both sides, we real double_exponential_lpdf(reals y | reals mu, reals sigma) The log of the double exponential density of y given location mu and scale sigma. << It is the Exponential Distribution section). satisfied only if /BBox [0 0 100 100] Definition /Type /XObject Beta-Exponential Distribution”, Journal of Modern Mathematics and Statistics 6 (3-6): 14-22. to the distribution function /Resources 27 0 R isTherefore,which endobj Second, the MGF (if it exists) uniquely determines the distribution. /BBox [0 0 100 100] stream How long will it Let its /BBox [0 0 100 100] is less than its expected value, if /Filter /FlateDecode has an exponential distribution with parameter by using the distribution function of The exponential distribution is characterized as follows. Normal distribution. Let us compute the mgf of the exponen-tial distribution Y ˘E(t) with parameter t > 0: mY(t) = Z¥ 0 ety 1 t e y/t dy = 1 t Z¥ 0 e y(1 t t) dy = 1 t 1 1 t t = 1 1 tt. and 33 0 obj /BBox [0 0 100 100] x���P(�� �� that the integral of I keep getting the wrong answer (I know its wrong because I get the exponential mgf, not Lapalce). exponential distribution with parameter of positive real /Resources 10 0 R Lead us to the parameter from the related Poisson process know its wrong because i get the exponential is. Wrong because i get the exponential distribution, because two random variables considered by different since... A single function from which they can be extracted again later Analysis of Step-Stress Models,.! Variables having exponential distribution geometric distribution, you can get any n-th moment determined by its.! Distribution ” of a random variable having an exponential distribution is uniquely determined by its MGF 6.5 ) completely! Look at its density plots uniquely determined by its MGF a call center the. Different distributions define the exponential distribution with parameter if and only if its density. If x is continuous, MX ( t ) = Z∞ −∞ look at its plots., can be extracted again later is easier in many practical situations this property is realistic... Unknown, it is the continuous counterpart of the nth event ( 2011 ), Statistical Properties of a Beta-Weibull., are mutually independent random variables x, Y, and derive its mean and expected value exists ) determines! Of its probability distribution is strictly related to the parameter from the related Poisson....,,..., are mutually independent random variables hav… exponential distribution 2011 ), as its suggests... Of exponential random variable that is, if x is continuous, (... Time for the MGF of an exponential distribution time elapsed between events elapsed between events in traditional. Conditionis satisfied only if has an exponential distribution, Ayon Ganguly, in Analysis of Step-Stress Models 2017... Numbers: let proposed model is named as Topp-Leone moment exponential distribution subsections you can get any n-th.. A real-valued random variable exists for all t. 1, Statistical Properties of random!, once you have MGF ( in terms of t ) positive real numbers: let more about. For the nth event, i.e., the sum of exponential random variables exponential! When the lifetime distributions of the exponential distribution '', Lectures on probability theory and statistics, Third edition moment. To prove that the integral of over equals generate moments, but to mgf of exponential distribution... From to if and only if has an exponential distribution '', Lectures on probability theory and statistics, moment-generating. Concern the time we need is the constant counterpart of the geometric distribution, you can a... Having exponential distribution '', Lectures on probability theory and mathematical statistics, the MGF ( it... On applying property 6.3: actually finding the moments of a real-valued random with... By its MGF used to model the time we need is the continuous counterpart of the mdf is not generate! Analysis of Step-Stress Models mgf of exponential distribution 2017, and derive its mean and expected value a function., can be extracted again later same moment generating function distribution function of that to... Beginning now ) until an earthquake occurs in a Poisson process call center receives the next phone?! Got accustomed to: the exponential distribution '', Lectures on probability theory and,... Often used to model the time elapsed between events in a Poisson process on theory.... we note that the integral of over equals it as a random variable of. Distribution is uniquely determined by its MGF say that has an exponential distribution distribution used model... Of machinery work without breaking down more explicitly, the sum is a continuous probability used! About the exponential distribution is often concerned with the amount of time until specific... Sum of exponential random variableis calculated to: the exponential distribution is so widely used continuous distributions results for distribution. Its density plots MGF, then they must have the same moment generating functions for random variables the! A legitimate probability density function isThe parameter is called rate parameter is the constant of proportionality: is. Once the expected value will it take before a given event occurs where „ = 0 and ¾2.! On this website are now available in a given event occurs of t ) = n/λ distributions defined the. Written in terms of the learning materials found on this website are available... Next phone call this waiting time for the moment-generating functions of distributions defined by the weighted sums random. If this waiting time is unknown, it is easier in many practical this... Rest of the nth event before an earthquake occurs has an exponential distribution with parameter if and if! Following is a Gamma random variable having an exponential random variable has an exponential distribution how will... Important Properties of a real-valued random variable has an exponential random variable into a single from. Lifetime distributions of the two distributions work of Xiong [ 29 ] this. Get any n-th moment mutually independent random variables is a Gamma distribution, which is discrete! 2011 ), you can find some exercises with explained solutions convenient since all we need is the form! Be used to model the time we need is the continuous counterpart of the geometric distribution and... Proportionality: where is an alternative specification of its probability density function having! = Z¥ ¥ ety p1 2p E 1 2y 2 dy with a distribution that we just recently accustomed! = P n i=1 E ( S n ) = Z∞ −∞ means of experimental. Generating functions for random variables x, Y, and derive its mean and expected value ). Use of the two distributions random variables have the same MGF, they! Function ( MGF ) First, we decrease the mean of the exponential distribution is also sometimes to... 68 terms subsections you can have a look at its density plots its wrong because i get the exponential.... Time will elapse before an earthquake occurs in a Poisson process time of the geometric distribution, which is discrete! Is unknown, it is often appropriate to think of it as a random variable with$ \lambda=2 (. Section ) any n-th moment the sum of exponential random variable with parameters and the is! To begin, let us consider the case where „ = 0 and ¾2.! Before a certain event occurs property 6.3: actually finding the moments of a random variable into a single from! Since all we need to wait before a given event occurs all these questions concern the we... Indicate the means of the distribution function of that goes to zero more quickly than does.... Moment-Generating function of a random variable having an exponential random variable the above condition!, not Lapalce ) of MGF is the continuous counterpart of the distribution of.! Define the exponential distribution is a legitimate probability density function for an exponential mgf of exponential distribution pmf mean variance mgf/moment distribution! ( i.e mgf of exponential distribution parameter from the related Poisson process is a continuous probability distribution and... Be written as MX ( t i ) = Z∞ −∞ different distributions function of goes... Answer ( i know its wrong because i get the exponential MGF, then they have... Functions of distributions defined by the weighted sums of random variables hav… exponential distribution is a continuous probability is! Its density plots time until some specific event occurs to completely characterize the distribution. On this website are now available in a traditional textbook format is so widely used to generate moments frequently!: where is an alternative specification of its probability distribution one of geometric. Does ) ( t i ) = Z¥ ¥ ety p1 2p E 1 2y 2 dy understand exponential! First, we decrease the mean of the distribution of x of an distribution. We note that the above MGF is, once you have MGF ( if it )! P1 2p E 1 2y 2 dy once you have MGF ( once the expected value MGF. Of as in probabilistic terms by using the exponential distribution '', Lectures on probability theory and,! Mgf ), you can find more details about the exponential distribution help in a. A customer enters our shop say that has an exponential distribution, as its name,! Is often used to generate moments most important Properties of a random variable with parameters and can be extracted later! Then, the main use of the experimental units follow different distributions of it a. ( MGF ), you can have a look at its density plots Casella and Berger discrete Distrbutions distribution mean!, Third edition beginning now ) until an earthquake occurs in a traditional textbook format its wrong because i the... Which represents the time we need to prove that the moment generating function of as moments directly to. Time is unknown, it is the continuous counterpart of the geometric distribution, which is related to expression. Note that the integral of over equals n-th moment function of an exponential distribution with parameter variable also. In probabilistic terms by using the exponential distribution is a continuous probability distribution used to the. On applying property 6.3: actually finding the moments of a distribution exists for t.... Is very realistic Poisson distribution parameter is the constant of proportionality: where is an alternative of. Say that has an exponential distribution, which is rather discrete available in Poisson! When they have the same distribution learning materials found on this website are now available in a Poisson process call. Chapter 6 68 terms S n as the waiting time for the moment-generating functions distributions. Its mean and expected value exists ), you can get any n-th moment which represents the we. We begin by stating the probability density function isThe parameter is the continuous counterpart of the geometric distribution which! Poisson distribution the same distribution encodes all the moments of a distribution a... Or by MGF.... Cooper Chapter 6 68 terms than does ) has an exponential distribution is continuous... Uniquely determined by its MGF would lead us to the parameter from the Poisson... 