0 X {\displaystyle S_{n}=\sum _{i=1}^{n}a_{i}X_{i}} ⁡ may not exist. 1 To get around this difficulty, we use some more advanced mathematical theory and calculus. α = n t a X Thus we obtain formulas for the moments of the random variable X: This means that if the moment generating function exists for a particular random variable, then we can find its mean and its variance in terms of derivatives of the moment generating function. Moment generating functions are positive and log-convex, with M(0) = 1. {\displaystyle M_{\alpha X+\beta }(t)=e^{\beta t}M_{X}(\alpha t)}, If The moment-generating function is so named because it can be used to find the moments of the distribution. X is a vector and ↦ ) μ ≤ One way to calculate the mean and variance of a probability distribution is to find the expected values of the random variables X and X2. t {\displaystyle e^{tX}} t X The following is a formal definition. α {\displaystyle M_{X}(-t)} {\displaystyle m_{i}} Weisstein, Eric W. "Moment-Generating Function." X ⟩ X {\displaystyle X} t k, (6.3.1) where m k = E[Yk] is the k-th moment of Y. Theorem for Characteristic Functions." + M , density function , if there exists for any wherever this expectation exists. 0 ( The end result is something that makes our calculations easier. X Join the initiative for modernizing math education. 0 However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. We let X be a discrete random variable. holds: since the PDF's two-sided Laplace transform is given as, and the moment-generating function's definition expands (by the law of the unconscious statistician) to. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. {\displaystyle M_{X}(t)} ) ) {\displaystyle x\mapsto e^{xt}} The lognormal distribution is an example of when this occurs. > X > t Section 3.5: Moments and Moment Generating Functions De–nition 1 Expected Values of integer powers of X and X are called moments. This statement is also called the Chernoff bound. This function allows us to calculate moments by simply taking derivatives. {\displaystyle a>0} Here are some examples of the moment-generating function and the characteristic function for comparison. Given a random variable and a probability {\displaystyle X} , we have. i t 0 If there is a positive real number r such that E(etX) exists and is finite for all t in the interval [-r, r], then we can define the moment generating function of X. Various lemmas, such as Hoeffding's lemma or Bennett's inequality provide bounds on the moment-generating function in the case of a zero-mean, bounded random variable. {\displaystyle M_{X}(t)=e^{t^{2}/2}} {\displaystyle t} X ( There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. This statement is not equivalent to the statement "if two distributions have the same moments, then they are identical at all points." ( always exists and is equal to 1. ) t and Practice online or make a printable study sheet. M , we can choose Moment generating functions possess a uniqueness property. function satisfies, If is differentiable at zero, then the By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead. a {\displaystyle f_{X}(x)} m ( What Is the Skewness of an Exponential Distribution? {\displaystyle m_{n}} {\displaystyle M_{X}(t)} n The Moment Generating Function of a Random Variable, Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra. exists. {\displaystyle i} ∑ For powers of X these are called moments about the mean. ≥ {\displaystyle Y} ) t {\displaystyle t>0} ( / of Statistics, Pt. X i ) is the Fourier transform of its probability density function The next example shows how the mgf of an exponential random variableis calculated. i , then 0 an such that. ) has moment generating function ) M = , {\displaystyle t>0} In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. X t X Explore anything with the first computational knowledge engine. i n th moments … This is consistent with the characteristic function of X X Since Unlimited random practice problems and answers with built-in Step-by-step solutions. M X moment-generating function. for , where denotes P As its name implies, the moment generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0. {\displaystyle \mathbf {X} } X The moment generating function has many features that connect to other topics in probability and mathematical statistics. : M x of Statistics, Pt. [2]. ⟨ 2, 2nd ed. where . ) and the two-sided Laplace transform of its probability density function when the moment generating function exists, as the characteristic function of a continuous random variable i In other words, we say that the moment generating function of X is given by: This expected value is the formula Σ etx f (x), where the summation is taken over all x in the sample space S. This can be a finite or infinite sum, depending upon the sample space being used. See the relation of the Fourier and Laplace transforms for further information. X , which is within a factor of 1+a of the exact value. t n Expected Value of a Binomial Distribution, Explore Maximum Likelihood Estimation Examples, How to Calculate Expected Value in Roulette, Maximum and Inflection Points of the Chi Square Distribution, How to Find the Inflection Points of a Normal Distribution, B.A., Mathematics, Physics, and Chemistry, Anderson University. 72-77, {\displaystyle f(x)} {\displaystyle tX} / . ( a Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. t ) {\displaystyle \mathbf {t} } ) M and recall that {\displaystyle f(x)} is a Wick rotation of its two-sided Laplace transform in the region of convergence. β ) f x e The mean is M’(0), and the variance is M’’(0) – [M’(0)]2. > X The mean and the variance of a random variable X with a binomial probability distribution can be difficult to calculate directly. where 2 {\displaystyle \mathbf {t} } The moment-generating function of a real-valued distribution does not always exist, unlike the characteristic function. The moment-generating function is the expectation of a function of the random variable, it can be written as: Note that for the case where The moment-generating function is so called because if it exists on an open interval around t = 0, then it is the exponential generating function of the moments of the probability distribution: That is, with n being a nonnegative integer, the nth moment about 0 is the nth derivative of the moment generating function, evaluated at t = 0. M instead of  β is the two-sided Laplace transform of X t th moment. is monotonically increasing for ( 2 1 A moment-generating function, or MGF, as its name implies, is a function used to find the moments of a given random variable. is a continuous random variable, the following relation between its moment-generating function Differentiating ) ", ThoughtCo uses cookies to provide you with a great user experience. In other words, the moment-generating function is the expectation of the random variable f By using ThoughtCo, you accept our, Use of the Moment Generating Function for the Binomial Distribution, How to Calculate the Variance of a Poisson Distribution. E((X )3) is called the third moment about the mean. , {\displaystyle \mu } e = e 1951. see Calculations of moments below. times with respect to In summary, we had to wade into some pretty high-powered mathematics, so some things were glossed over. {\displaystyle X} m {\displaystyle P(X\geq a)\leq e^{-a^{2}/2}} X Knowledge-based programming for everyone. is the is the dot product. For example, when X is a standard normal distribution and 2, 2nd ed. X {\displaystyle \alpha X+\beta } Upper bounding the moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X. is. T t e when the latter exists. Moment generating functions can be used to calculate moments of X. If the moment generating functions for two random variables match one another, then the probability mass functions must be the same. ⋅ and setting {\displaystyle n} ) , t and any a, provided Kenney, J. F. and Keeping, E. S. "Moment-Generating and Characteristic Functions," "Some Examples of Moment-Generating Functions," and "Uniqueness {\displaystyle M_{X}(t)} f {\displaystyle M_{X}(t)}