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MODERN PROBABILITY
THEORY AND ITS
APPLICATIONS
EMANUEL PARZEN
CONTENTS
CHAPTER
1
PAGE
PROBABILITY THEORY AS THE: STUDY OI' MATHEMATICAL MODELS
OP KANDOM PHENOMENA
1
Probability theory as the study of random phenomena
2
Probability theory as the study of mathematical models of random phenomena
3
The sample description space of a random phenomenon
4
Events
5
The definition of probability as a function of events on a simple description space
6
Finite sample description spaces
7
Finite sample description spaces with equally likely descriptions
8
Notes on the literature of probability theory
1
1
5
8
11
17
23
25
28
2
BASIC PROBABILITY THEOKY
1
Samples and n-tuples
2
Posing probability problems mathematically
3
The number of "successes" in a simple
4
Conditional probability
5
Unordered and partitioned simples‑occupancy problems
6
The probability of occurrence of a given number of events
32
32
42
51
60
67
76
3
INDEPENDENCE AND DEPENDENCE
1
Independent events and families of events
2
Independent trials
3
Independent Bernoulli trials
4
Dependent trials
5
Markov dependent Bernoulli trials
6
Markov chains
87
87
94
100
113
128
136
4
NUMERICAL‑VALUED RANDOM PHENOMENA
1
The notion of a numerical‑valued random phenomenon
2
Specifying the probability law of a numerical‑valued random phenomenon
Appendix: The evaluation of integrals and sums
3
Distribution functions
4
Probability laws
5
The uniform probability law
6
The normal distribution and density functions
148
148
151
160
166
176
184
188
6
7
The normal distribution and density functions
Numerical n‑tuple valued random phenomena
188
193
5
MEAN AND VARIANCE OF A PROBABILITY LAW
1
The notion of an average
2
Expectation of a function with respect to a probability law
3
Moment‑generating functions
4
Chebyshev's inequality
5
The law of large numbers for independent repeated Bernoulli trials
6
More about expectation
199
199
203
215
225
228
232
6
NORMAL, POISSON, AND RELATED PROBABILITY LAWS
1
The importance of the normal probability law
2
The approximation of the binomial probability law by the
normal and Poisson probability laws
3
The Poisson probability law
4
The exponential and gamma probability laws
5
Birth and death processes
237
237
RANDOM VARIABLES
1
The notion of a random variable
2
Describing a random variable
3
An example, treated from the point of view of numerical
n‑tuple valued random phenomena
4
The same example treated from the point of view of random variables
5
Jointly distributed random variables
6
Independent random variables
7
Random samples, randomly chosen points (geometrical probability),
and random division of an interval
8
The probability law of a function of a random variable
9
The probability law of a function of random variables
10
The joint probability law of functions of random variables
11
Conditional probability of an event given a random variable.
Conditional distributions
268
268
270
8
EXPECTATION OF A RANDOM VARIABLE
1
Expectation, mean, and variance of a random variable
2
Expectations of jointly distributed random variables
3
Uncorrelated and independent random variables
4
Expectations of sums of random variables
5
The law of large numbers and the central limit theorem
6
The measurement signal‑to‑noise ratio of a random variable
7
Conditional expectation. Best linear prediction
343
343
354
361
366
371
378
384
9
SUMS OF INDEPENDENT RANDOM VARIABLES
1
The problem of addition of independent random variables
2
The characteristic function of a random variable
3
The characteristic function of a random variable specifies its probability law
4
Solution of the problem of the addition of independent
random variables by the method of characteristic functions
5
Proofs of the inversion formulas for characteristic functions
391
391
394
400
SEQUENCES OF RANDOM VARIABLES
1
Modes of convergence of a sequence of random variables
2
The law of large numbers
3
Convergence in distribution of a sequence of random variables
4
The central limit theorem
5
Proofs of theorems concerning convergence in distribution
414
414
417
424
430
434
7
10
239
251
260
264
276
282
285
294
298
308
316
329
334
405
408
Tables
441
Answers to Odd‑Numbered Exercises
447
Index
459
LIST OF IMPORTANT TABLES
TABLE
2-6A
PAGE
THE PROBABILITIES OF VARIOUS EVENTS DEFINED ON THE GENERAL
OCCUPANCY AND SAMPLING PROBLEMS
84
5-3A
5-3B
6-6A
I
SOME FREQUENTLY ENCOUNTERED DISCRETE PROBABILITY LAWS
AND THEIR MOMENTS AND GENERATING FUNCTIONS
218
SOME FREQUENTLY ENCOUNTERED CONTINUOUS PROBABILITY
LAWS AND THEIR MOMENTS AND GENERATING FUNCTIONS
220
MEASUREMENT SIGNAL TO NOISE RATIO OF RANDOM VARIABLES
OBEYING VARIOUS PROBABILITY LAWS
380
AREA UNDER THE NORMAL DENSITY FONCTION; A TABLE OF
441
(x) =
II
III
TOP
BINOMIAL PROBABILITIES; A TABLE OF
n = 1,2,…,10 AND VARIOUS VALUES OF p
px (1 – p) n-x , FOR
POISSON PROBABILITIES; A TABLE OF eVALUES OF A
x ,/x!, FOR VARIOUS
442
444