Moment-generating functionals

Table 2.5-2 Mean Variance and Moment-Generating Functions for Several Distributions ...

Mutual information is thus a random variable since it is a real valued function defined on the points of an ensemble. Consequently, it has an average, variance, distribution function, and moment generating function. It is important to note that mutual information has been defined only on product ensembles, and only as a function of two events, x and y, which are sample points in the two ensembles of which the product ensemble is formed. Mutual information is sometimes defined as a function of any two events in an ensemble, but in this case it is not a random variable. It should also be noted that the mutual... 

Bounds on the Distribution of Sums of Random Variables. Let be a random variable, assuming the value Z. with probability Pr(zk) for 1 h K. Define the moment generating function of x as... 

Thus, although Pr(z) is generally quite difficult to calculate for large JV, the moment generating function can be expressed quite simply as... 

Recall now that the letters in xx are chosen independently with the probability distribution p = (Pi, , > ) and when xx is sent the output is governed by the transition probabilities Pr( i). Thus, each of the terms d( ln,pn) in Eq. (4-123) is an independent random variable with the moment generating function... 

Now consider N pairs of random variables each having the distribution above, and each pair being statistically independent of all other pairs. Define w = 2 -t wn> = 2 -i and define HN(r,t) to be the joint moment generating function of w,z. 

The moment generating function, when it exists, is defined by... 

The characteristic function and the moment generating function are important tools for computing moments of distributions, studying limits of sequences of distributions, and finding the distribution of sums of independent variables. If Z — X + F, where X and T are independently distributed according to the distribution functions F(x) and 6( y) respectively, the distribution function of Z is given by... 

To obtain the variance we use the characteristic function approach (of course one can also use the moment generating function). 

Mintzer, David, 1 Mitropolsky, Y. A361,362 Mixed groups, 727 Modality of distribution, 123 Models in operations research, 251 Modification, method of, 67 Molecular chaos, assumption of, 17 Miller wave operator, 600 Moment generating function, 269 Moment, 119 nth central, 120... 

The moment generating function M(t) = (e,x) is particularly useful to compute moments. Expanding the exponential, we get... 

Calculate the moment generating function of the general normal distribution f(x) given by... 

From the definition of the moment generating function, we write... 

The mean of the gamma distribution is cufl and its variance a/ 2. The moment generating function is... 

No simple form of the moment generating function exists. In the special case where 0C =a2 = 1, the beta distribution reduces to the uniform distribution over [0, 13- Finally, we will frequently refer to Snedecor s F-distribution. A random variable defined over ]0, + 00 [ is distributed with the F-distribution with v, and v2 degrees of freedom... 

Another way of handling changes of variables is through the moment generating function. If Z is the sum of two independent random variables X and Y, integration of the two variables under the integral can be carried out independently, hence... 

Consequently, the distribution of the sum Z of two normal variables X and Y with respective moment generating functions... 

Likewise, the distribution of the sum Z of two gamma variables X and Y with identical second parameter ft and with moment generating functions Mx(t)=( 1 —fk) ax and My(t)=(l—f t) ar has interesting additive properties. Again... 

The coefficients of the multivariable Taylor series expansion of G J) about the point where the Schwinger probes vanish are elements of the ROMs. Thus G J) is known as the generating functional for ROMs. Mathematically, the RDMs of the functional G J) are known as the moments. The moment-generating functional G(y) may be used to define another functional W J), known as the cumulant-generating functional, by the relation... 

This is an analogue of the classical moments-generating functional discussed by Kubo [39]. Upon expanding the exponential as a power series, the operator J f acts to place each term in so-called normal order, in which all creation operators are to the left of all annihilahon operators j/. By virtue of this ordering (and only by virtue of this ordering). 

Moment Generating Function. For the random variable X. with probability density function/(x), if the function M(t) = E[ea] exists, it is the moment generating function. Assuming the function exists, it can be shown that d M(l)/df t=0 = E x. Find the moment generating functions for... 

The sum is the sum of probabilities for a Poisson distribution with parameter Ax , which equals 1, so the term before the summation sign is the moment generating function, M(t) = cxp X(e -1)]. 

Moment generating function for a sum of variables. When it exists, the moment generating function has a one to one correspondence with the distribution. Thus, for example, if we begin with some random variable and find that a transformation of it has a particular MGF, we may infer that the function of the random variable has the distribution associated with that MGF. A useful application is the following ... 

Using the same approach as in part b., it follows that the moment generating function for a sum of random variables with means u, and standard deviations a, is... 

It is also the moment generating function in the sense that the coefficients of its Taylor expansion in k are the moments ... 

In chapter I it was shown how the handling of moments and cumulants was facilitated by the use of a moment generating function. A similar tool will now be introduced with respect to the / . Instead of the auxiliary variable k we now need an auxiliary function, or test function v(t and instead of a generating function we have therefore a functional, i.e., a quantity depending on all the values that v takes for — oo < t < oo (indicated by [v]). The generating functional for the / is... 

This shows that log L is the generating functional of the gm, just as the cumulants were generated by the logarithm of the moment generating function. Of course, one may define the gm by (5.7) and then prove that they obey (5.1). 

The generalization of the concept of a characteristic function to stochastic processes is the characteristic functional (In a different connection this idea was used in section II.3.) Let Y(t) be a given random process. Introduce an arbitrary auxiliary test function k(t). Then the characteristic or moment generating functional is defined as the following functional of k(t ... 

---