The exponential distribution is characterized by means of the distribution of a weighted sum of independent, identically distributed random variables. However, its form is not mathematically convenient, as it requires the. I have read that the sum of gamma random variables with the same scale parameter is another gamma random variable. Sum of exponential random variables towards data science. B on the exact distribution of a normalized ratio of the weighted. We show that the decay is subexponential, and identify the rate function in. Sum of n variables generate n random variate y i s and sum. The random variable can be one of the independent exponential random variables such that is with probability with. Weighted sums of subexponential random variables and their maxima. The target of this study is the randomly weighted sum s.
Van leeuwaarden and temme 2011 attempted to determine good approximation of the distribution of. The conditions considered turn out to be sufficient in the case of two random variables only. This means that the large deviation event is already triggered by one of the terms in the sum being large. The random variable, with and being independent exponentially distributed random variables with mean one, is considered. Analysis of bit error probability for imperfect timing. Limiting behavior of weighted sums of independent random variables chow, y. Nagaraja the ohio state university columbus oh, usa abstract. Jul 15, 20 we consider the distribution of the sum and the maximum of a collection of independent exponentially distributed random variables. The probability density function pdf of a positive definite quadratic form in central or noncentral normal variables can be represented as a series expansion in a number of different ways. In addition, we also give the relevant precise asymptotics.
The main problem is estimating the parameter that has the main state in applicable research. Properties in this section we study the different properties of the we. A comparison of efficient approximations for a weighted sum. Many situations arise where a random variable can be defined in terms of the sum of other random variables. The fact that the means and variances add when summing s. We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. Suppose x1 and x2 are two independent and identically distributed random variables, with the probability density function fy y and cumulative distribution function cdf fy y. Characterizations of the exponential distribution by.
Say x is an exponential random variable of parameter. Almost certain summability of independent, identically distributed random variables chow, y. The target of this study is the randomly weighted sum s n xn i1 ix i. In this paper we give new exponential inequalities for weighted sums of realvalued independent random variables bounded on the right. The most important of these situations is the estimation of a population mean from a sample mean. Sums of discrete random variables 289 for certain special distributions it is possible to. Weighted sums of subexponential random variables and. Sum of two independent exponential random variablesedit. Simply knowing that the result is gaussian, though, is enough to allow one to predict the parameters of the density. A new estimate of the probability density function pdf of the sum of a random number of independent and identically distributed iid random variables is shown. Order statistics from independent exponential random. A random variable x or its distribution f satisfying fx 0 for any x 2. The pdf of weighted sum of exponential random variables. Large deviations for weighted sums of stretched exponential random.
For sums of two variables, pdf of x convolution of pdfs of y 1 and y 2. Pdf on sep 6, 2015, anming dong and others published logarithmic expectation of the sum of exponential random variables for wireless communication performance evaluation find, read and cite. The erlang distribution, the hypoexponential distribution and the hyperexponential distribution are special cases of phasetype distributions that are. If a random variable x has this distribution, we write x exp. On the estimation of parameter of weighted sums of. On the distribution of sums of independent exponential random. Large deviations for infinite weighted sums of stretched. An estimate of the probability density function of the sum of. The exponential inequality for weighted sums of a class of linearly negative quadrant dependent random variables is established, which extends and improves the corresponding ones obtained by ko et al.
An estimate of the probability density function of the sum. Hypoexponential distribution the distribution of a general sum of exponential random variables. Exponential inequalities for weighted sums of bounded random variables rio, emmanuel, electronic communications in probability, 2015. A connection between the pdf and a representation of the convolution characteristic function as a. Distribution of a sum of weighted central chisquare variables. Some stochastic inequalities for weighted sums arxiv. Our results are extensions of the results of bennett 1962 to weighted sums. The sum pdf is represented as a sum of normal pdfs weighted according to the pdf. The problem of inversion of the mgfchf of a sum of lognormals to obtain the cdf pdf is considered with special reference to methods related to post widder technique, gaussian quadrature and the fourier series method. The analytical model is verified by numerical simulations. Large deviations for weighted sums of stretched exponential. The simplest way to generate we random deviate is to use the stochastic representation 5,i.
Let x be a continuous random variable with range a. Note that the center is at 0,0 so that the variables can have both negative and positive values. Let xi exponential inequality for weighted sums of a class of. This section deals with determining the behavior of the sum from the properties of the individual components. Pdf large deviations for weighted sums of stretched. Variance of the weighted sum of two random variables. Weighted sums of certain dependent random variables. For example, 1, 2, n could be a sample corresponding to the random variable x. Exponential inequalities for weighted sums of bounded. Sum of independent exponential random variables paolo.
On the sum of exponentially distributed random variables. One of the most important classes of heavytailed distributions is the subexponential class s. In order to present our large deviation result for weighted sums of stretched exponential random variables, we will use slightly di. In this section we consider only sums of discrete random variables. Distribution of a sum of weighted noncentral chisquare. The focus is laid on the explicit form of the density functions pdf of noni. We consider the probability that a weighted sum of n i. Pdf on sep 6, 2015, anming dong and others published logarithmic expectation of the sum of exponential random variables for wireless communication performance evaluation. Hyperexponential distribution the distribution whose density is a weighted sum of exponential densities. The exponential distribution exhibits infinite divisibility. Pdf logarithmic expectation of the sum of exponential. The probability distribution function pdf of a sum of two independent random variables is the.
Improved approximation of the sum of random vectors by the skew normal distribution christiansen, marcus c. Weighted sums of subexponential random variables and their. Request pdf stochastic inequalities for weighted sum of two random variables independently and identically distributed as exponential let x and y be two random variables which are. Download scientific diagram the pdf of weighted sum of exponential random variables. Randomly weighted sums of subexponential random variables. Exponential random variables and the sum of the top order statistics h. Sums of a random variables 47 4 sums of random variables many of the variables dealt with in physics can be expressed as a sum of other variables. Jan 18, 2014 in order to present our large deviation result for weighted sums of stretched exponential random variables, we will use slightly di. The hyperexponential and hypoexponential distributions. Those are recovered in a simple and direct way based on conditioning. Stochastic inequalities for weighted sum of two random. Aug 16, 2019 the notation means that the random variable takes the particular value is a random variable and capital letters are used. Sum of two independent exponential random variables. On the estimation of parameter of weighted sums of exponential.
Such a random variable is said to follow a hyperexponential distribution. X1 and x2 are independent exponential random variables with the rate x1exp. Tail probability of randomly weighted sums of dependent. F distribution, other examples of subexponential distributions include the. Suppose that x and y are statistically independent normal random variables, with f xx 1. The notation means that the random variable takes the particular value is a random variable and capital letters are used. Sumofindependentexponentials university of bristol. The probability density function pdf of an exponential distribution is. This density can not be described using densities of standard r. Keywords large deviations weighted sums subexponential random variables selfnormalized weights quenched and annealed large deviations random projections kernels nonparametric regression rights this work is licensed under a creative commons attribution 3. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The method based on the complex exponential integral which makes use of the derivative. In this paper we show that estimating the parameter by using the. We consider the distribution of the sum and the maximum of a collection of independent exponentially distributed random variables.
A new class of weighted exponential distributions rameshwar d. Thus, p m l1 is a sum of weighted noncentral chisquare random variables. Majorization allows us to compare tail probabilities of weighted sums of exponential variates, when the weight coefficients are composed of interior and boundary points of a unit circle. Methods for calculating the probability distribution of sums. Ive also seen the paper by moschopoulos describing a method for the summation of a general set of gamma random variables.
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