File Name: moment generating function and their properties .zip
A generating function of a real-valued random variable is an expected value of a certain transformation of the random variable involving another deterministic variable. Most generating functions share four important properties:. Property 1 is perhaps the most important.
In probability theory and statistics , the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables.
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A generating function of a real-valued random variable is an expected value of a certain transformation of the random variable involving another deterministic variable. Most generating functions share four important properties:. Property 1 is perhaps the most important. Often a random variable is shown to have a certain distribution by showing that the generating function has a certain form.
The process of recovering the distribution from the generating function is known as inversion. Property 2 is frequently used to determine the distribution of a sum of independent variables. By contrast, recall that the probability density function of a sum of independent variables is the convolution of the individual density functions, a much more complicated operation.
Property 3 is useful because often computing moments from the generating function is easier than computing the moments directly from the probability density function. The last property is known as the continuity theorem. Often it is easer to show the convergence of the generating functions than to prove convergence of the distributions directly.
The numerical value of the generating function at a particular value of the free variable is of no interest, and so generating functions can seem rather unintuitive at first.
But the important point is that the generating function as a whole encodes all of the information in the probability distribution in a very useful way. Generating functions are important and valuable tools in probability, as they are in other areas of mathematics, from combinatorics to differential equations. We will study the three generating functions in the list below, which correspond to increasing levels of generality. The fist is the most restrictive, but also by far the simplest, since the theory reduces to basic facts about power series that you will remember from calculus.
The third is the most general and the one for which the theory is most complete and elegant, but it also requires basic knowledge of complex analysis. The one in the middle is perhaps the one most commonly used, and suffices for most distributions in applied probability. We will also study the characteristic function for multivariate distributions, although analogous results hold for the other two types.
In the basic theory below, be sure to try the proofs yourself before reading the ones in the text. The probability generating function can be written nicely in terms of the probability density function. The expansion follows from the discrete change of variables theorem for expected value. Recall from calculus that a power series can be differentiated term by term, just like a polynomial. Each derivative series has the same radius of convergence as the original series but may behave differently at the endpoints of the interval of convergence.
This is a standard result from the theory of power series. The factorial moments can be computed from the derivatives of the probability generating function. The factorial moments, in turn, determine the ordinary moments about 0 sometimes referred to as raw moments. Recall that the expected product of independent variables is the product of the expected values. Our next generating function is defined more generally, so in this discussion we assume that the random variables are real-valued.
This follows from the change of variables theorem for expected value. The Laplace transform is named for Pierre Simon Laplace , and is widely used in many areas of applied mathematics, particularly differential equations. For more details see the advanced section on properties of the integral in the chapter on Distributions.
Thus, the derivatives of the moment generating function at 0 determine the moments of the variable hence the name. In the language of combinatorics, the moment generating function is the exponential generating function of the sequence of moments. Thus, a random variable that does not have finite moments of all orders cannot have a finite moment generating function.
Even when a random variable does have moments of all orders, the moment generating function may not exist. A counterexample is constructed below. For nonnegative random variables which are very common in applications , the domain where the moment generating function is finite is easy to understand. Of course, there are complementary results for non-positive random variables, but such variables are much less common.
Next we consider what happens to the moment generating function under some simple transformations of the random variables. Location-scale transformations frequently arise when units are changed, such as length changed from inches to centimeters or temperature from degrees Fahrenheit to degrees Celsius. The probability generating function of a variable can easily be converted into the moment generating function of the variable.
The following theorem gives the Chernoff bounds , named for the mathematician Herman Chernoff. These are upper bounds on the tail events of a random variable.
Our last generating function is the nicest from a mathematical point of view. Once again, we assume that our random variables are real-valued. Many of the properties of the characteristic function are more elegant than the corresponding properties of the probability or moment generating functions, because the characteristic function always exists.
This follows from the change of variables theorem for expected value, albeit a complex version. The Fourier transform is named for Joseph Fourier , and is widely used in many areas of applied mathematics. As with other generating functions, the characteristic function completely determines the distribution.
Indeed, the general inversion formula given next is a formula for computing certain combinations of probabilities from the characteristic function. A special inversion formula holds for continuous distributions:. This formula is essentially the inverse Fourrier transform. Moreover, this can be done even when only some of the moments exist. Next we consider how the characteristic function is changed under some simple transformations of the variables.
The characteristic function of a random variable can be obtained from the moment generating function, under the basic existence condition that we saw earlier. The final important property of characteristic functions that we will discuss relates to convergence in distribution.
Since we are only concerned with distributions, the random variables need not be defined on the same probability space. There are analogous versions of the continuity theorem for probability generating functions and moment generating functions. The continuity theorem can be used to prove the central limit theorem , one of the fundamental theorems of probability. All of the generating functions that we have discussed have multivariate extensions. However, we will discuss the extension only for the characteristic function, the most important and versatile of the generating functions.
There are analogous results for the other generating functions. The marginal characteristic functions and the characteristic function of the sum can be easily obtained from the joint characteristic function:.
Naturally, the results for bivariate characteristic functions have analogies in the general multivariate case. Only the notation is more complicated. As always, be sure to try the computational problems yourself before expanding the solutions and answers in the text.
Suppose that an ace-six flat die and a flat die are rolled. Use probability generating functions to find the probability density function of the sum of the scores.
Two fair, 6-sided dice are rolled. Use probability generating functions to find the probability density function of the sum of the scores, and identify the distribution.
The process is named for Jacob Bernoulli. A separate chapter on the Bernoulli Trials explores this process in more detail. This follows immediately from the PGF of an indicator variable and the result for sums of independent variables. The Poisson distribution is named after Simeon Poisson and is widely used to model the number of random points in a region of time or space; the parameter is proportional to the size of the region of time or space.
The Poisson distribution is studied in more detail in the chapter on the Poisson Process. The Poisson family of distributions is closed with respect to sums of independent variables, a very important property. The right distribution function of the Poisson distribution does not have a simple, closed-form expression. The following exercise gives an upper bound. Substituting gives the upper bound.
The following theorem gives an important convergence result that is explored in more detail in the chapter on the Poisson process. This distribution is widely used to model failure times and other random times, and in particular governs the time between arrivals in the Poisson model. The exponential distribution is studied in more detail in the chapter on the Poisson Process. This follows from the previous result and the result for sums of independent variables.
The Erlang distribution is a special case of the gamma distribution and is studied in more detail in the chapter on the Poisson Process. Continuous uniform distributions arise in geometric probability and a variety of other applied problems. Compute each of the following:. So the normal family of distributions in closed under location-scale transformations. The family is also closed with respect to sums of independent variables:. The Pareto distribution is named for Vilfredo Pareto.
It is a heavy-tailed distribution that is widely used to model financial variables such as income. The Pareto distribution is studied in more detail in the chapter on Special Distributions.
However, the characteristic function of the Pareto distribution does not have a simple, closed form. The Cauch distribution is studied in more generality in the chapter on Special Distributions.
The proof of this result requires contour integrals in the complex plane, and is given in the section on the Cauchy distribution in the chapter on special distributions. For the Pareto distribution, only some of the moments are finite; so course, the moment generating function cannot be finite in an interval about 0. We will now give an example of a distribution for which all of the moments are finite, yet still the moment generating function is not finite in any interval about 0.
Furthermore, we will see two different distributions that have the same moments of all orders. The lognormal distribution is studied in more generality in the chapter on Special Distributions. This distribution has finite moments of all orders, but infinite moment generating function.
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. Specifically, I can understand that in real world, from data, we can get an estimation of the probability distribution. If we cannot get it, where does it come from? If it is from the Laplace transform of pdf, i.
The expected value and variance of a random variable are actually special cases of a more general class of numerical characteristics for random variables given by moments. Note that the expected value of a random variable is given by the first moment , i. 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. Theorem 3. We also find the variance.
In this chapter, we discuss the theory necessary to find the distribution of a transformation of one or more random variables. While the emphasis of this text is on simulation and approximate techniques, understanding the theory and being able to find exact distributions is important for further study in probability and statistics. For one, it is a part of the common knowledge base that probabilists and statisticians know and use when discussing things. We may need to apply theory to know what types of things to approximate or compute when faced with a new situation. Even if we are only concerned in approximate solutions in known contexts, we can often get much better approximations if we apply some theory to the problem. We will see examples of this below. One method that is often applicable is to compute the cdf of the transformed random variable, and if required, take the derivative to find the pdf.
Here, after formally defining the gamma distribution we haven't done that yet?! But the p. The gamma p.
Sign in. The moments are the expected values of X, e. The first moment is E X ,. They are important characteristics of X. The mean is the average value and the variance is how spread out the distribution is. But there must be other features as well that also define the distribution.
In the paper, some properties related to the moment generating function of a fuzzy variable are discussed based on uncertainty theory. And we obtain the result that the convergence of moment generating functions to an moment generating function implies convergence of credibility distribution functions. Thats, the moment generating function characterizes a credibility distribution. Request Permissions.
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