Mathematics
Sum of Independent Random Variables
The sum of independent random variables refers to the result of adding together two or more random variables that are statistically independent of each other. In probability theory, the sum of independent random variables has a well-defined distribution, and its properties can be analyzed using techniques such as convolution. This concept is fundamental in understanding the behavior of random processes and systems.
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9 Key excerpts on "Sum of Independent Random Variables"
eBook - PDFProbability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers
- Roy D. Yates, David J. Goodman(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
9 Sums of Random Variables Random variables of the form W n = X 1 + · · · + X n (9.1) appear repeatedly in probability theory and applications. We could in principle derive the probability model of W n from the PMF or PDF of X 1 , . . . , X n . However, in many practical applications, the nature of the analysis or the properties of the random variables allow us to apply techniques that are simpler than analyzing a general n-dimensional probability model. In Section 9.1 we consider applications in which our interest is confined to expected values related to W n , rather than a complete model of W n . Subsequent sections emphasize techniques that apply when X 1 , . . . , X n are mutually independent. A useful way to analyze the Sum of Independent Random Variables is to transform the PDF or PMF of each random variable to a moment generating function. The central limit theorem reveals a fascinating property of the sum of indepen- dent random variables. It states that the CDF of the sum converges to a Gaussian CDF as the number of terms grows without limit. This theorem allows us to use the properties of Gaussian random variables to obtain accurate estimates of prob- abilities associated with sums of other random variables. In many cases exact calculation of these probabilities is extremely difficult. 9.1 Expected Values of Sums The expected value of a sum of any random variables is the sum of the expected values. The variance of the sum of any random variable is the sum of all the covariances. The variance of the Sum of Independent Random Variables is the sum of the variances. 289 290 CHAPTER 9 SUMS OF RANDOM VARIABLES The theorems of Section 5.7 can be generalized in a straightforward manner to describe expected values and variances of sums of more than two random variables. Theorem 9.1 For any set of random variables X 1 , . . . , X n , the sum W n = X 1 + · · · + X n has expected value E [W n ] = E [X 1 ] + E [X 2 ] + · · · + E [X n ] .
eBook - PDFProbability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers
- Roy D. Yates, David J. Goodman(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
9 Sums of Random Variables Random variables of the form W n = X 1 + · · · + X n (9.1) appear repeatedly in probability theory and applications. We could in principle derive the probability model of W n from the PMF or PDF of X 1 , . . . , X n . However, in many practical applications, the nature of the analysis or the properties of the random variables allow us to apply techniques that are simpler than analyzing a general n-dimensional probability model. In Section 9.1 we consider applications in which our interest is confined to expected values related to W n , rather than a complete model of W n . Subsequent sections emphasize techniques that apply when X 1 , . . . , X n are mutually independent. A useful way to analyze the Sum of Independent Random Variables is to transform the PDF or PMF of each random variable to a moment generating function. The central limit theorem reveals a fascinating property of the sum of indepen- dent random variables. It states that the CDF of the sum converges to a Gaussian CDF as the number of terms grows without limit. This theorem allows us to use the properties of Gaussian random variables to obtain accurate estimates of prob- abilities associated with sums of other random variables. In many cases exact calculation of these probabilities is extremely difficult. 9.1 Expected Values of Sums The expected value of a sum of any random variables is the sum of the expected values. The variance of the sum of any random variable is the sum of all the covariances. The variance of the Sum of Independent Random Variables is the sum of the variances. 306 9.1 EXPECTED VALUES OF SUMS 307 The theorems of Section 5.7 can be generalized in a straightforward manner to describe expected values and variances of sums of more than two random variables. Theorem 9.1 For any set of random variables X 1 , . . . , X n , the sum W n = X 1 + · · · + X n has expected value E [W n ] = E [X 1 ] + E [X 2 ] + · · · + E [X n ] .
eBook - PDFRandom Vibrations
Analysis of Structural and Mechanical Systems
- Loren D. Lutes, Shahram Sarkani(Authors)
- 2004(Publication Date)
- Butterworth-Heinemann(Publisher)
Thus, the description of a random variable is simply a description of its probabilities. It should perhaps be noted that there is nothing to be gained by debating whether a certain physical quantity truly is a random variable. The more pertinent question is whether our uncertainty about the value of the quantity can be usefully modeled by a random variable. As in all other areas of applied mathematics, it is safe to assume that our mathematical model is never identical to a physical quantity, but that does not necessarily preclude our using the model to obtain meaningful results. As presented in Section 2.2, probabilities are always defined for sets of possible outcomes, called events. For a problem described by a single real random variable X , any event of interest is always equivalent to the event of X belonging to some union (finite or infinite) of disjoint intervals of the real line. In some problems we are most interested in events of the type X u = , in which u is a particular real number. In order to include this situation within the idea of events corresponding to intervals, we can consider the event to be X I u ∈ with I u u u = [ , ] being a closed interval that includes only the single point u . In other problems, we are more interested in intervals of finite length. Probably the most general way to describe the probabilities associated with a given random variable is with the use of the cumulative distribution function, which will be written as F X ( ) ⋅ . The argument of this function is always a real number, and the domain of definition of the function is the entire real line. That is, for any real random variable the argument of the cumulative distribution function can be any real number. The definition of the F X ( ) ⋅ function is in terms of a probability of X being smaller than or equal to a 1 Later we will also use complex random variables and vector random variables, but these are unnecessary at this stage of the development.
eBook - PDF- G P Beaumont(Author)
- 2005(Publication Date)
- Woodhead Publishing(Publisher)
270 Sums of Random Variables [Ch.14 1 = 1 ησ a η (14.11) The standard deviation of X, a/s/n, is also known as the standard error of the mean. As η increases, V{JC) decreases, hence the larger the sample drawn the more likely it is that the sample mean is near the population mean. More precisely, by Tchebychev's inequaUty, equation (10.5), V(X) a' ?iiX-p>k)<'i^ = — . (14.12) k' nk' Since a'/nk' tends to zero as« tends to infinity, the limit of the Ρτ(Χ — μ> k) also tends to zero. We also have a crude estimate of this probability given n. Results for covariances similar to equation (14 .3) for variances may also be obtained. Thus CiaX, +bX2.cYi +dY,) = E{[aXi +bX2 -EiaXi + bX^yicY, + dY, -EicY, + dY,)]) = E{[aXi -aE{Xt) + bX2 -bE(X2)][cY, -c £(y,) + dY2 -dEiY2)]) = E{ac[Xi -Ε(Χχ)][Υχ -E(Y,)] + ad[X, -Ε{Χχ)][Υ2-E{Y2)] + bc[X2 -Ε{Χ2)]ΙΥ, -Ε{ΥΟ] + + bd[X2 -Ε{Χ2)][Υ2-Ε{Υ2)]} = acCiX, ,Υ,) + adC(Xi , ^2) + bcC(X2 .Υι) + bdCiX2, Y2 )· (14 .13) The mode of expansion for the covariance of two linear combinations of random variables is evident from (14 .13). Example 2 iXi, X2 are independent, (a) C(Xi.Xi +X2) = CiXi.Xi)+C{Xi,X2)=V{X,) + 0. (b) C{3Xi - 4X2 , 2Xi +5X2) = Ci3Xi ,2Xi) + C (3Ar,, 5 ^2) + C(-4 ^2 ,2Xi) + Ci-4X2 , 5 ^2) = 6CiX,.Xi ) + 0 + 0 -20CiX2.X2) = 6K(Z,)-20F (X2). Sec. 14.2] 271 F(X) = σ' This result will apply in sampling a finite population without replacement. 14.2 SUMS OF INDEPENDENT RANDOM VARIABLES, DISTRIBUTIONS The task of this section is to consider ways of evaluating the distribution of X, + Xl where X,, Xj are independent random variables with known distribu-tions. The first method to be considered does not require the existence of any moments and hence is very general, though sometimes tedious to apply. Theorem 14.1. If X,, Xi are independent discrete random variables with p.d.f.s /i ( ^ 1 ) .
eBook - PDF- Vladimir I. Rotar(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 3 Discrete Random Variables 1,2 1 RANDOM VARIABLES AND VECTORS 1.1 Random variables and their distributions Loosely put, a random variable (r.v.) is a variable whose values are determined “by chance.” More precisely, the value of a r.v. depends on which elementary outcome has occurred when the experiment under consideration has been performed. For example, when tossing a coin twice, we may be interested merely in the number of heads, no matter in which order they appeared. Possible results are given in the table below. Outcomes The number of heads ω 1 = { HH } 2 ω 2 = { HT } 1 ω 3 = { TH } 1 ω 4 = { TT } 0 So, while there are four outcomes, the number of heads may take on three values: either 0, or 1, or 2. We also see that the number of heads is completely determined by the outcome occurred. In the general case, when considering an experiment with a sample space Ω = { ω } , we de fi ne a random variable as a function X ( ω ) on the space Ω . EXAMPLE 1. In the case of rolling two dice, the sample space Ω = { ω } consists of 36 outcomes. The r.v. X ( ω ) , the sum of the numbers on the dice, may assume all integer values from 2 to 12. EXAMPLE 2. We choose at random a point from the disk { ( x , y ) : x 2 + y 2 ≤ 1 } ; see Example 1 .1.1-7. An elementary outcome ω may be identified with a point ( x , y ) from the disk. The distance from the point chosen to the center of the disk is the r.v. X ( ω ) = x 2 + y 2 which may take on any value from the interval [ 0 , 1 ] . As a rule, to simplify the notation, we will write X for X ( ω ) . A r.v. is said to be discrete if it can take on only a finite or countable number of possible values. If it is not the case, we call the r.v. non-discrete . Another term in use is a continuous r.v. , but it is slightly misleading; see Example 6 below. EXAMPLE 3. The number of successes in n trials is a discrete r.v.
eBook - PDF- Joseph W. Goodman(Author)
- 2015(Publication Date)
- Wiley(Publisher)
The theory of probability is based on these axioms. The problem of assigning specific numerical values to probabilities of various events is not addressed by the axiomatic approach, but rather is left to our physical intuition. Whatever number we assign for the probability of a given event must agree with our intuitive feeling for the limiting relative frequency of that event. In the end, we are simply building a statistical model that we hope will represent the experiment. The necessity to hypothesize a model should not be disturbing, for every determinis- tic analysis likewise requires hypotheses about the physical entities involved and the transformations they undergo. Our statistical model must be judged on the basis of its accuracy in describing the behavior of experimental results over many trials. We are now prepared to introduce the concept of a random variable. To every possible elementary event A of our underlying random experiment, we assign a real number u(A). The random variable U consists of all possible u(A), together with an associated measure of their probabilities. Note especially that the random vari- able consists of both the set of possible values and their associated probabilities, and hence, it encompasses the entire statistical model that we hypothesize for the random phenomenon. 8 RANDOM VARIABLES 2.2 DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS A random variable U is called discrete if the possible experimental outcomes consist of a discrete set of numbers. A random variable is called continuous if the possi- ble experimental results can lie anywhere on a continuum of values. Occasionally, a mixed random variable is encountered, with possible outcomes that lie on both a discrete set (with certain probabilities) and a continuum.
eBook - PDF- Kathleen Subrahmaniam(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Describing the Joint Behavior of Several Random Variables In Chapter 6 we discussed how two or more random variables might be defined on the same sample space. We summarized their joint behavior by a joint probability function» At times the probability function may be a cum- bersome description of the random variation« As a remedy we developed more concise measures—the mean and the variance. In this chapter we will introduce some concise measures involving more than one variable: covari-ance and correlation coefficient. We will also develop laws of expectation and variance involving several variables. 8.1 EXPECTATION OF A FUNCTION OF TWO RANDOM VARIABLES Extending the concepts of the last chapter to several variables, we will now define the expected value of a function of two random variables» DEFINITION 8 .1 Let X and Y be two random variables with joint prob-ability function p(Xj, yj) for i = 1, 2, ».., n and j = 1, 2, ». . , m. Consider a function G(X, Y). Then the expected value of G(X, Y) is n m E[G(X, Y)] = £ E G(x , y )p(x y ) . ___________________ i=l 3=1 _______________________________ __ Note that this is just a generalization of Definition 7.2. To illustrate this definition, consider the following example. Example 1: Let X and Y have the joint p.f. 8 139 140 Chapter 8 X y 0 1 2 3 1 1 l « o 0 -8 8 1 1 „ 2 o ô ô o 8 8 2 2 3 o il< il< o 8 8 What is the expected value of (a) XY? (b) X /Y ? Solution: (a) Here G(X, Y) =XY. Then E(XY) = ( 0 )( 1 )| + (3)(1)| + (1)(2)|+ (2)(2)|+ (1)(3)|+ (2)(3)| = y . (b) Similarly, Definition 8 .1 can easily be extended for more than two random variables. 8.2 COVARIANCE In studying the behavior of two random variables simultaneously, we are faced with trying to find a measure of how the r . v . ’s vary jointly. Essen-tially we are interested in generalizing the concept of 'Variance.
- John A. Gubner(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
Again, since this union is disjoint, the result is immediate. Since µ depends on X , if more than one random variable is under discussion, we write µ X ( B ) instead. We thus see that different random variables induce different probability mea-sures on IR. Another term for measure is distribution . Hence, we call µ X the distribution of X . More generally, the term “distribution” refers to how probability is spread out. As we will see later, for discrete random variables, once we know the probability mass function, we can compute µ X ( B ) = P ( X ∈ B ) for all B of interest. Similarly, for the continuous ran-dom variables of Chapter 4, once we know the probability density function, we can compute µ X ( B ) = P ( X ∈ B ) for all B of interest. In this sense, probability mass functions, probabil-ity density functions, and distributions are just different ways of describing how to compute P ( X ∈ B ) . 2.2: Discrete random variables Note 2. To derive (2.4), we apply the law of total probability as given in (1.27) with A = { X ∈ B } and B i = { X = x i } . Since the x i are distinct, the B i are disjoint, and (1.27) says that P ( X ∈ B ) = ∑ i P ( { X ∈ B }∩{ X = x i } ) . (2.33) Now observe that if x i ∈ B , then X = x i implies X ∈ B , and so { X = x i } ⊂ { X ∈ B } . This monotonicity tells us that { X ∈ B }∩{ X = x i } = { X = x i } . On the other hand, if x i / ∈ B , then we cannot have X = x i and X ∈ B at the same time; in other words { X ∈ B }∩{ X = x i } = ∅ . 98 Introduction to discrete random variables It now follows that P ( { X ∈ B }∩{ X = x i } ) = P ( X = x i ) , x i ∈ B , 0 , x i / ∈ B . Substituting this in (2.33) yields (2.4). 2.3: Multiple random variables Note 3. In light of Note 1 above, we do not require that (2.5) hold for all sets B and C , but only for all Borel sets B and C . Note 4. If X 1 ,..., X n are independent, we show that any subset of them must also be independent.
eBook - PDFProbability Theory
A First Course in Probability Theory and Statistics
- Werner Linde(Author)
- 2016(Publication Date)
- De Gruyter(Publisher)
We investigate now the sum of two independent chi-squared distributed random variables. Recall Definition 1.6.26: A random variable X is 7 2 n -distributed if it is A 2, n 2 - distributed. Hence, Proposition 4.6.11 implies the following result. Proposition 4.6.16. Suppose that X is 7 2 n -distributed and that Y is 7 2 m -distributed for some n, m ≥ 1. If X and Y are independent, then X + Y is 7 2 n+m -distributed. Proof: Because of Proposition 4.6.11, the sum X + Y is A 2, n 2 + m 2 = 7 2 n+m -distributed. This proves the assertion. ∎ Proposition 4.6.16 has the following important consequence. Proposition 4.6.17. Let X 1 , . . . , X n be a sequence of independent N (0, 1)-distributed random variables. Then X 2 1 + ⋅ ⋅ ⋅ + X 2 n is 7 2 n -distributed. Proof: Proposition 4.1.5 asserts that the random variables X 2 j are 7 2 1 -distributed. Fur- thermore, because of Proposition 4.1.9 they are also independent. Thus a successive application of Proposition 4.6.16 proves the assertion. ∎ Our next and final aim in this section is to investigate the distribution of the sum of two independent normally distributed random variables. Here the following important result is valid. Proposition 4.6.18. Let X 1 and X 2 be two independent random variables distributed according to N (, 1 , 3 2 1 ) and N (, 2 , 3 2 2 ). Then X 1 + X 2 is N (, 1 + , 2 , 3 2 1 + 3 2 2 )-distributed. Proof: In a first step we treat a special case, namely , 1 = , 2 = 0 and 3 1 = 1. To simplify the notation, set + = 3 2 . Thus we have to prove the following: if X 1 and X 2 are N (0, 1)- and N (0, + 2 )-distributed, then X 1 + X 2 is N (0, 1 + + 2 )-distributed. Let p 0,1 and p 0,+ 2 be the corresponding densities introduced in eq. (1.47). Then we have to prove that p 0,1 ⋆ p 0,+ 2 = p 0,1++ 2 . (4.35) 188 4 Operations on Random Variables To verify this start with (p 0,1 ⋆ p 0,+ 2 )(x) = 1 20 + ∞ –∞ e –(x–y) 2 /2 e –y 2 /2+ 2 dy = 1 20 + ∞ –∞ e – 1 2 (x 2 –2xy+(1++ –2 )y 2 ) dy .
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