... X({) = 1, 3, 5, 7, . . .}. Find P(A). Ans. 3 CHAP. 21 RANDOM VARIABLES 2.4 DISCRETE RANDOMVARIABLESANDPROBABILITY MASS FUNCTIONS A. Definition : Let X be a r.v. with cdf ... digits 1 and 0 randomly with probabilities 0.6 and 0.4, respectively. (a) What is the probability that two 1s and three 0s will occur in a five-digit sequence? (b) What is the probability ... and y = r sin 9 (that is, using polar coordinates), we have Thus, and 2.22. Consider a function Find the value of a such that f (x) is a pdf of a continuous r.v. X. RANDOM VARIABLES...
... A, B, C are said to be independent if and only if (1 SO) Schaum's Outline ofTheory and Problems of Probability, Random Variables, andRandom Processes Hwei P. Hsu, Ph.D.Professor ... digits 1 and 0 randomly with probabilities 0.6 and 0.4, respectively. (a) What is the probability that two 1s and three 0s will occur in a five-digit sequence? (b) What is the probability ... 2 and property 3. Since A c S, we have and by Eq. (2.31), RANDOM VARIABLES [CHAP 2 2.30. Find the mean and variance of the r.v. X of Prob. 2.20. From Prob. 2.20, the pdf...
... 00 U Bi = U A, for all n 2 1, and U B, = U A, = A, i=l i=l i=l i=l CHAP. 21 RANDOM VARIABLES 2.4 DISCRETE RANDOMVARIABLESANDPROBABILITY MASS FUNCTIONS A. Definition ... digits 1 and 0 randomly with probabilities 0.6 and 0.4, respectively. (a) What is the probability that two 1s and three 0s will occur in a five-digit sequence? (b) What is the probability ... called the probability mass function (pmf) of the discrete r.v. X. Properties of pdx) : The cdf FX(x) of a discrete r.v. X can be obtained by 2.5 CONTINUOUS RANDOMVARIABLESAND PROBABILITY...
... Large Numbers and the Central Limit Theorem 128 Solved Problems 129 Chapter 5. RandomProcesses 161 5.1 Introduction 161 5.2 RandomProcesses 161 5.3 Characterization of RandomProcesses 161 ... of RandomProcesses 162 5.5 Discrete-Parameter Markov Chains 165 5.6 Poisson Processes 169 5.7 Wiener Processes 172 Solved Problems 172 Chapter 6. Analysis and Processing of RandomProcesses ... Theorems 122 4.1 Introduction 122 4.2 Functions of One Random Variable 122 4.3 Functions of Two RandomVariables 123 4.4 Functions of n RandomVariables 124 4.5 Expectation 125 4.6 Moment Generating...
... ofrandom variables, called circular complex random variables. Circularity is a type of symmetry in the distributions ofthe real and imaginary parts of complex randomvariablesand stochastic processes, ... the randomvariables themselves are complex: the χ2, F, and β distributionsall describe real randomvariables functionally dependent on complex Gaussians.Let z and q be independent scalar random ... Leon-Garcia, A., Probability andRandomProcesses for Electrical Engineering,2nd ed.,Addison-Wesley, Reading, MA, 1994.[4] Melsa, J. and Sage, A.,An Introduction to Probabilityand Stochastic Processes, Prentice-Hall,Englewood...
... ρ-mixing random variablesand to present some results on complete convergence under some suitableconditions. Some results generalize previous known results for rowwise independent random variables. 1. ... generated by {Xi; m ≤ i ≤ n}.The ρ-mixing randomvariables were first introduced by Kolmogorov and Rozanov1. The limiting behavior of ρ-mixing randomvariables is very rich, for example, these ... Theory of Probabilityand Its Applications, vol. 2, pp. 222–227, 1960.2 I. A. Ibragimov, “A note on the central limit theorem for dependent random variables, ” Theory of Probability and Its Applications,...
... for Positively Associated Random Variablesand ApplicationsGuodong Xing,1Shanchao Yang,2 and Ailin Liu31Department of Mathematics, Hunan University of Science and Engineering, Yongzhou,425100 ... associated variables, ” Statistics & Probability Letters,vol. 73, no. 2, pp. 189–197, 2005.5 S C. Yang a nd M. Chen, “Exponential inequalities for associated randomvariablesand strong ... inequalities. Shao and Yu2 generalized later the previous results. Recently, Ioannides and Roussas 3 establishedaBernstein-Hoeffding-type inequality for stationary and positively associated random vari-ables...
... inequality for acceptable randomvariables generalizes and improvesthe corresponding results presented by Yang for NA randomvariablesand Wang etal. for NOD random variables. Using the exponential ... results of Yang [9] for NA random variablesand Wang et al. [10] for NOD random variables. In Section 3, we willstudy the complete convergence for acceptable randomvariables using the exponentialinequalities ... acceptable random variables. MSC(2000): 60E15, 60F15.Keywords: acceptable random variables; exponential inequality; complete conver-gence.1 IntroductionLet {Xn, n ≥ 1} be a sequence of random variables...
... presented by Yang for NA randomvariablesand Wang et al.for NOD random variables. Using the exponential inequalities, we further study thecomplete convergence for acceptable random variables. MSC(2000): ... acceptable random variables. For example, Xing et al. [6] consider a strictlystationary NA sequence of random variables. According to the sentence above, asequence of strictly stationary and NA random ... results ofYang [9] for NA randomvariablesand Wang et al. [10] for NOD random v ariables. InSection 3, we will study the complete convergence for acceptable random variables using the exponential...
... covering probability, random variables, and stochastic processes is [353].2.1 PROBABILITY DISTRIBUTIONS AND DENSITIES2.1.1 Distribution of a random variableIn this book, we assume that randomvariables ... random vectorsy and z, and (2.57) still holds. A similarargument applies to the random vectorsy and z.Example 2.6 First consider the random variables x and ydiscussed in Examples 2.2 and ... the random vectorx(n)=x(n)x(n 1)::: x(n m)]T(2.118) 2 Random Vectors and IndependenceIn this chapter, we review central concepts of probability theory,statistics, and random processes. ...
... Transformations based on both cross spectral matrix and covariance matrix branches to analysis and identification of multi-variate random pressure fields. The random pressure fields are determined due ... eigenvalue and associated spectral mode are clarified in reconstructing the random pressure fields and identifying the hidden physical phenomena inside this pressure fields. Keywords: random pressure ... distribution and correlation of random fluctuating pressure field on surface of structural section. The fluctuating pressure field can be represented as spatially-correlated multi-variate random processes. ...
... composed random variable η =νk=1ξk, where ξ1, ξ2, are independent identically distributed randomvariablesand ν is a positive value random, independent of all ξk.In [1] and [2], ... distribution of the composed randomvariablesand their stabilitiesDotor thesis, Hanoi 2000.[2] Tran Kim Thanh, Nguyen Huu Bao, On the geometric composed variablesand the estimate of the stable ... Physics 23 (2007) 70-74On the stability of the distribution function of the composed random variables by their index random variableNguyen Huu Bao∗Faculty of Infomation Technology, Water Resources...
... required parameters (L and p) of OMP-PKS+RS and DCS-SOMP in Section 4.2 and (2) the performance evaluation in Section 4.3. 4.2. Evaluation for L and p Both OMP-PKS+RS and DCS-SOMP require ... 1c and the vectorization by lexicographic ordering, respectively. (a) Girl, (b) Jelly Beans, (c) Airplane (F-16), and (d) Mandrill. Figure 3. The ensemble of compressed measurement vector and ... reconstruction examples when L and p were set according to Tables 1 and 2, respectively. The top and the bottom rows of each image in Figure 12 show the reconstruction based on DCS-SOMP and OMP-PKS+RS, respectively....