Probability and Random Processes

[1322/100%]

G. R. Grimmett, D. R. Stirzaker

Chapter 1Events and their probabilities[79/100%]

Section 1.1Introduction

Section 1.2Events as sets[5/100%]

Section 1.3Probability[7/100%]

Section 1.4Conditional probability[8/100%]

Section 1.5Independence[10/100%]

Section 1.6Completeness and product spaces

Section 1.7Worked examples[6/100%]

Section 1.8Problems[43/100%]

Chapter 2Random variables and their distributions[45/100%]

Section 2.1Random variables[6/100%]

Section 2.2The law of averages[3/100%]

Section 2.3Discrete and continuous variables[5/100%]

Section 2.4Worked examples[2/100%]

Section 2.5Random vectors[6/100%]

Section 2.6Monte Carlo simulation

Section 2.7Problems[23/100%]

Chapter 3Discrete random variables[144/99%]

Section 3.1Probability mass functions[5/100%]

Section 3.2Independence[5/100%]

Section 3.3Expectation[12/100%]

Section 3.4Indicators and matching[14/100%]

Section 3.5Examples of discrete variables[5/100%]

Section 3.6Dependence[11/100%]

Section 3.7Conditional distributions and conditional expectation[14/100%]

Section 3.8Sums of random variables[8/100%]

Section 3.9Simple random walk[8/88%]

Section 3.10Random walk: counting sample paths[5/100%]

Section 3.11Problems[57/100%]

Chapter 4Continuous random variables[248/100%]

Section 4.1Probability density functions[4/100%]

Section 4.2Independence[7/100%]

Section 4.3Expectation[13/100%]

Section 4.4Examples of continuous variables[15/100%]

Section 4.5Dependence[20/100%]

Section 4.6Conditional distributions and conditional expectation[15/100%]

Section 4.7Functions of random variables[27/100%]

Section 4.8Sums of random variables[11/100%]

Section 4.9Multivariate normal distribution[11/100%]

Section 4.10Distributions arising from the normal distribution[7/100%]

Section 4.11Sampling from a distribution[18/100%]

Section 4.12Coupling and Poisson approximation[8/100%]

Section 4.13Geometrical probability[18/100%]

Section 4.14Problems[74/100%]

Chapter 5Generating functions and their applications[166/100%]

Section 5.1Generating functions[12/100%]

Section 5.2Some applications[13/100%]

Section 5.3Random walk[9/100%]

Section 5.4Branching processes[8/100%]

Section 5.5Age-dependent branching processes[2/100%]

Section 5.6Expectation revisited[7/100%]

Section 5.7Characteristic functions[13/100%]

Section 5.8Examples of characteristic functions[17/100%]

Section 5.9Inversion and continuity theorems[9/100%]

Section 5.10Two limit theorems[15/100%]

Section 5.11Large deviations[5/100%]

Section 5.12Problems[56/100%]

Chapter 6Markov chains[190/100%]

Section 6.1Markov processes[13/100%]

Section 6.2Classification of states[7/100%]

Section 6.3Classification of chains[10/100%]

Section 6.4Stationary distributions and the limit theorem[17/100%]

Section 6.5Reversibility[12/100%]

Section 6.6Chains with finitely many states[10/100%]

Section 6.7Branching processes revisited[4/100%]

Section 6.8Birth processes and the Poisson process[12/100%]

Section 6.9Continuous-time Markov chains[14/100%]

Section 6.10Kolmogorov equations and the limit theorem[5/100%]

Section 6.11Birth–death processes and imbedding[9/100%]

Section 6.12Special processes[4/100%]

Section 6.13Spatial Poisson processes[10/100%]

Section 6.14Markov chain Monte Carlo[6/100%]

Section 6.15Problems[57/100%]

Chapter 7Convergence of random variables[105/100%]

Section 7.1Introduction[5/100%]

Section 7.2Modes of convergence[12/100%]

Section 7.3Some ancillary results[14/100%]

Section 7.4Laws of large numbers[5/100%]

Section 7.5The strong law[3/100%]

Section 7.6The law of the iterated logarithm[1/100%]

Section 7.7Martingales[5/100%]

Section 7.8Martingale convergence theorem[3/100%]

Section 7.9Prediction and conditional expectation[9/100%]

Section 7.10Uniform integrability[7/100%]

Section 7.11Problems[41/100%]

Chapter 8Random processes[49/98%]

Section 8.1Introduction

Section 8.2Stationary processes[4/100%]

Section 8.3Renewal processes[5/100%]

Section 8.4Queues[5/100%]

Section 8.5The Wiener process[8/100%]

Section 8.6Lévy processes and subordinators[7/100%]

Section 8.7Self-similarity and stability[6/100%]

Section 8.8Time changes[4/100%]

Section 8.9Existence of processes

Section 8.10Problems[10/90%]

Chapter 9Stationary processes[50/100%]

Section 9.1Introduction[6/100%]

Section 9.2Linear prediction[2/100%]

Section 9.3Autocovariances and spectra[7/100%]

Section 9.4Stochastic integration and the spectral representation[3/100%]

Section 9.5The ergodic theorem[3/100%]

Section 9.6Gaussian processes[6/100%]

Section 9.7Problems[23/100%]

Chapter 10Renewals[48/100%]

Section 10.1The renewal equation[7/100%]

Section 10.2Limit theorems[5/100%]

Section 10.3Excess life[5/100%]

Section 10.4Applications[3/100%]

Section 10.5Renewal–reward processes[7/100%]

Section 10.6Problems[21/100%]

Chapter 11Queues[46/100%]

Section 11.1Single-server queues

Section 11.2M/M/1[7/100%]

Section 11.3M/G/1[4/100%]

Section 11.4G/M/1[3/100%]

Section 11.5G/G/1[3/100%]

Section 11.6Heavy traffic[1/100%]

Section 11.7Networks of queues[7/100%]

Section 11.8Problems[21/100%]

Chapter 12Martingales[73/100%]

Section 12.1Introduction[9/100%]

Section 12.2Martingale differences and Hoeffding's inequality[4/100%]

Section 12.3Crossings and convergence[6/100%]

Section 12.4Stopping times[8/100%]

Section 12.5Optional stopping[10/100%]

Section 12.6The maximal inequality[1/100%]

Section 12.7Backward martingales and continuous-time martingales[8/100%]

Section 12.8Some examples

Section 12.9Problems[27/100%]

Chapter 13Diffusion processes[79/100%]

Section 13.1Introduction

Section 13.2Brownian motion[1/100%]

Section 13.3Diffusion processes[11/100%]

Section 13.4First passage times[9/100%]

Section 13.5Barriers[3/100%]

Section 13.6Excursions and the Brownian bridge[5/100%]

Section 13.7Stochastic calculus[5/100%]

Section 13.8The Itô integral[6/100%]

Section 13.9Itô's formula[6/100%]

Section 13.10Option pricing[5/100%]

Section 13.11Passage probabilities and potentials[3/100%]

Section 13.12Problems[25/100%]