...
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 of
Theory 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 of
the real and imaginary parts of complex randomvariablesand stochastic processes, ... the randomvariables themselves are complex: the χ
2
, F, and β distributions
all 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...
... inequality for acceptable randomvariables generalizes and improves
the corresponding results presented by Yang for NA randomvariablesand Wang et
al. 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 will
study the complete convergence for acceptable randomvariables using the exponential
inequalities ... acceptable random variables.
MSC(2000): 60E15, 60F15.
Keywords: acceptable random variables; exponential inequality; complete conver-
gence.
1 Introduction
Let {X
n
, 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 the
complete convergence for acceptable random variables.
MSC(2000): ... acceptable random variables. For example, Xing et al. [6] consider a strictly
stationary NA sequence of random variables. According to the sentence above, a
sequence of strictly stationary and NA random ... results of
Yang [9] for NA randomvariablesand Wang et al. [10] for NOD random v ariables. In
Section 3, we will study the complete convergence for acceptable random variables
using the exponential...
... ρ-mixing
random variablesand to present some results on complete convergence under some suitable
conditions. Some results generalize previous known results for rowwise independent random
variables.
1. ... generated by {X
i
; m ≤ i ≤ n}.
The ρ-mixing randomvariables were first introduced by Kolmogorov and Rozanov
1. 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 Applications
Guodong Xing,
1
Shanchao Yang,
2
and Ailin Liu
3
1
Department 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 Yu
2 generalized later the previous results. Recently, Ioannides and Roussas 3 established
aBernstein-Hoeffding-type inequality for stationary and positively associated random vari-
ables...