Sum of random variables

Thus, the self information of a sequence of N symbols from a discrete memoryless source is the sum of N independent random variables, namely the self informations of the individual symbols. An immediate consequence of this is that the entropy of the sequence, ff(Ujr), is the average value of a sum of random variables so that... 

Finally, observe from Eq. (4-112) that D(Xlfy) and D(xm,y) are both defined as sums of random variables, and thus, using Eqs. (4-116) and (4-117) the problem of bounding Pe has been reduced to the problem of bounding the tails of the distributions of sums of random variables. This is best done by the Chernov bound technique, briefly described in the following paragraphs. For a more detailed exposition, see Fano,16 Chapter 8. 

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... 

Let SN( = X + + XNt be a sum of random variables with random subscript Nt independently and identically distributed (according to F(x)). Thus N is also a random variable, and it depends on t. We wish to compute the expected value and the variance of SNt. We have... 

Convolution The mathematical operation that finds the distribution of a sum of random variables from the distributions of its addends. The term can be generalized to refer to differences, products, quotients, etc. It can also be generalized to refer to intervals, p-boxes and Dempster-Shafer structures as well as distributions. 

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... 

P. Levy, Theorie de I addition des variables aleatoires, Gauthier-Villars, Paris, 1954 Calcul des probabilites, Gauthier-Villars, Paris, 1925 B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Random Variables, Addison-Wesley, Reading, 1954. 

B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Random Variables Addison-Wesley, Reading MA, 1954. 

The general form of this expression, from the variance of a random-sized sum of random variables, can be found in Example 4 in Appendix E of Kulkarni (1995, pp. 577-578). 

Variance of the sum of random variables. One can write for any choice of two random variables that... 

In a first step, the random parameter field is discretized and replaced by a finite sum of random variables. Assume that a suitable approximation is given by a linear combination of continuous and independent random variables i([Pg.3475]

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