F-distribution, 104
-measurable, 61
t-distribution, 104
Abel’s lemma, 55
absolute continuity, 66
absorbing
a. barrier, 173, 178, 223, 225
absorption probability, 221
adjacency, 217
aperiodic state, 228
arithmetic/geometric means, 124
auxiliary equation, 252
Banach matchbox problem, 36
Banach–Kuratowski theorem, 6
Bernoulli distribution, 26
pgf, 53
Bertrand’s paradox, 76
beta distribution, 69
bilateral/double exponential distribution, 68, 79
binomial
b. coefficient, 250
b. distribution, 26
pgf, 53
b.–Poisson limit theorem, 28, 150
birth process
simple b. p., 190
birth–death
b.–d. process, 193
birth–death–immigration process, 202
bivariate normal distribution, 100
Bonferroni’s inequality, 20
Boole’s inequality, 19
bounded convergence theorem, 234
extinction of b. p., 163
Buffon
B.’s needle, 77
B.’s noodle, 79
B.–Laplace needle, 80
Cantor
C. set, 109
cardinality, 7
Cauchy distribution, 69
characteristic function, 125
mean, 75
mgf, 118
moments, 111
Cauchy–Schwarz inequality, 116
central limit theorem, 139, 149
centre of gravity, 31
Chapman–Kolmogorov equations, 208
characteristic function, 119, 125
Chebyshev’s inequality, 137
Chevalier de Méré, 11
chi-squared distribution, 69
class property, 215
closed class, 212
combination, 250
communicating
c. classes, 212
c. states, 212
complementary solution, 254
conditional
c. pdf, 96
c. probability, 11
continuity
c. of probability measures, 16
continuous random variable, 66
expectation of c. r. v., 73
convergence
c. in distribution, 146
c. in mean square, 135
c. in probability, 136
weak c., 146
convergence theorem for Markov chains, 235
convex function, 122
correlation coefficient, 115
countable set, 6
coupling game, 237
coupon-collecting problem, 36, 49, 59, 131
covariance, 114
Cramér’s theorem, 142
de Méré’s paradox, 11
degree, 244
density function, 66
conditional pdf, 96
joint pdf, 86
detailed balance equations, 241
difference equation, 252
auxiliary equation, 252
complementary solution, 254
particular solution, 254
difference set, 5
Dirichlet pgf, 56
discrete
d. random variable, 23
expectation of d. r. v., 30
d. sample space, 9
distribution, 206
F-d., 104
t-d., 104
steady-state d., 231
Bernoulli d., 26
beta d., 69
bilateral/double exponential d., 68, 79
binomial d., 26
bivariate normal d., 100
Cauchy d., 69
chi-squared d., 69
equilibrium d., 231
extreme-value d., 68
Gaussian d., 69
geometric d., 26
invariant d., 231
log-normal d., 112
negative binomial d., 27
normal d., 69
Poisson d., 26
stationary d., 231
steady-state d., 199
tail of d., 121
distribution function, 62
joint d. f., 83
marginal d. f., 84
doubly stochastic matrix, 208
Ehrenfest dog–flea model, 242
equiprobable outcomes, 9
ergodicity, 228
erratic
e. bishops, 248
e. kings, 246
e. knights, 245
event, 4
decreasing sequence, 17
dependence, 12
elementary e., 4
increasing sequence, 16
independence, 12
pairwise independence, 13
event space, 5
expected value, 30
experiment, 3
exponential distribution, 64, 69
characteristic function, 125
lack-of-memory property, 103, 187
mgf, 118
extended
e. binomial theorem, 251
e. Markov property, 208
extinction
e. probability, 163
e. probability theorem, 163
e./survival theorem, 164
extreme-value distribution, 68, 201
false positives, 15
Fenchel–Legendre tranform, 143
first come, first served, 196
first-passage
probability, 214
time, 214
Fubini’s theorem, 86
functions of random variables, 71, 93
Gambler’s Ruin Problem, 173, 223, 248
gambling, 167
Gaussian distribution, 69
generating function, 50
Dirichlet g. f., 56
moment g. f., 117
probability g. f., 52
geometric distribution, 26
lack-of-memory property, 36
geometrical probability, 76
harmonic mean, 124
hitting
h. probability, 221
h. time, 221
image, 24
immigration–death process, 202
inclusion–exclusion formula, 21, 46
independence, 12, 41, 84, 88, 98
indicator function, 25, 44, 45
inequality
arithmetic/geometric mean i., 124
Bonferroni’s i., 20
Boole’s i., 19
Cauchy–Schwarz i., 116
Chebyshev’s i., 137
Jensen’s i., 122
Lyapunov’s i., 133
Markov’s i., 121
inter-arrival time, 187
invariant distribution, 231
inversion theorem, 128
irreducible chain, 212
Jacobian formula, 93
Jensen’s inequality, 122
joint
j. continuity, 85
j. distribution function, 83
j. mgf, 131
j. pdf, 86
j. pmf, 38
Kronecker delta, 214
lack-of-memory property
of exponential distribution, 103, 187
of geometric distribution, 36
Landau’s notation, 127
large deviation theorem, 144
law of large numbers, 135, 137, 148
law of the subconscious statistician, 31, 73
Lebesgue decomposition theorem, 110
linearity of expectation, 40, 97
log-normal distribution, 112
Lyapunov’s inequality, 133
marginal
m. distribution function, 84
m. pdf, 88
m. pmf, 39
Markov chain, 205
convergence theorem for M. c., 235
homogeneous M. c., 205
initial distribution, 206
reversibility in equilibrium, 241, 247
reversible M. c., 241
transition matrix, 206
transition probabilities, 208
Markov property, 205
Markov’s inequality, 121
mass function, 24
joint m. f., 38
matching, 21
matrix
doubly stochastic m., 208
stochastic m., 206
transition m., 206
mean, 30
mean recurrence time, 228
measurability, 61
mgf, 117
moment generating function, 117
joint mgf, 131
negative binomial distribution, 27
pgf, 53
nomad, 158
normal distribution, 69
characteristic function, 126
mgf, 118
moments, 74
standard n. d., 151
normal number, 20
null state, 228
order statistics, 151
Pólya’s theorem, 218
paradox
Bertrand’s p., 76
de Méré’s p., 11
Simpson’s p., 19
particular solution, 254
partition, 14
pdf, 66
joint pdf, 86
periodicity, 228
permutation, 250
persistence, 214
pgf, 52
Planet Zog, 105
pmf, 24
Poisson distribution, 26
pgf, 53
Poisson process, 182, 183, 206
compound P. p., 201
doubly stochastic P. p., 201, 204
inhomogeneous P. p., 201
inter-arrival time, 187
superposition, 183
thinned P. p., 183
positive state, 228
power set, 4
probability density function, 66
conditional pdf, 96
joint pdf, 86
marginal pdf, 88
pdf of product and ratio, 102
probability generating function, 52
Dirichlet pgf, 56
probability mass function, 24
joint pmf, 38
marginal pmf, 39
probability measure, 6
continuity of p. m., 16
countable additivity of p. m., 6, 22
finite additivity of p. m., 6
pure birth process, 190
queue discipline, 196
queueing, 195
random integers, 9
random process, 157
random sum formula, 57
random variable, 61
continuous r. v., 66
image of r. v., 24
independence, 41
joint continuity of r. v.s, 85
standardized r. v., 139
uncorrelated r.v.s, 117
random walk, 167, 206, 217, 248
recurrence/transience, 170, 218
reflected r. w., 249
simple r. w., 167
symmetric r. w., 167
transition probabilities, 168
rate function, 201
r. time, 228
mean r. t., 228
retaining barrier, 178, 199, 242, 243
reversible Markov chain, 241, 247
Riemann zeta function, 258, 259
sample space, 4
discrete s. s., 9
Simpson’s paradox, 19
singular distribution, 110
standard deviation, 114
state, 205
absorbing s., 212
aperiodic s., 228
ergodic s., 228
null s., 228
persistent s., 214
positive s., 228
s. space, 205
statistical sampling, 141
stochastic
s. matrix, 206
s. process, 157
stopping time, 224
strong Markov property, 188, 225
supporting tangent theorem, 123
symmetric difference, 5
tail, 121
theorem of total probability, 14
time reversal, 240
traffic intensity, 199
transition
t. matrix, 206
t. probabilities, 208
trial, 3
uncorrelated, 117
mean, 74
uniqueness theorem
u. t. for characteristic functions, 127
u. t. for mgfs, 120
u. t. for moments, 112
u. t. for pgfs, 52
Venn diagram, 8
weak convergence, 146
Weierstrass approximation theorem, 152