One Thousand Exercises in Probability

Cover......Page p0001.djvu
Title page......Page filename5.djvu
Preface......Page filename5_0003.djvu
Contents......Page filename5_0005.djvu
1.3 Exercises. Probability......Page filename5_0009.djvu
1.4 Exercises. Conditional probability......Page filename5_0010.djvu
1.5 Exercises. Independence......Page filename5_0011.djvu
1.8 Solutions to problems......Page filename5_0012.djvu
2.2 Exercises. The law of averages......Page filename5_0018.djvu
2.4 Exercises. Worked examples......Page filename5_0019.djvu
2.7 Solutions to problems......Page filename5_0020.djvu
3.2 Exercises. Independence......Page filename5_0024.djvu
3.3 Exercises. Expectation......Page filename5_0025.djvu
3.4 Exercises. Indicators and matching......Page filename5_0026.djvu
3.6 Exercises. Dependence......Page filename5_0027.djvu
3.7 Exercises. Conditional distributions and conditional expectation......Page filename5_0028.djvu
3.8 Exercises. Sums of random variables......Page filename5_0029.djvu
3.9 Exercises. Simple random walk......Page filename5_0030.djvu
3.11 Solutions to problems......Page filename5_0031.djvu
4.2 Exercises. Independence......Page filename5_0037.djvu
4.4 Exercises. Examples of continuous variables......Page filename5_0038.djvu
4.5 Exercises. Dependence......Page filename5_0039.djvu
4.6 Exercises. Conditional distributions and conditional expectation......Page filename5_0040.djvu
4.7 Exercises. Functions of random variables......Page filename5_0041.djvu
4.8 Exercises. Sums of random variables......Page filename5_0042.djvu
4.9 Exercises. Multivariate normal distribution......Page filename5_0043.djvu
4.11 Exercises. Sampling from a distribution......Page filename5_0044.djvu
4.12 Exercises. Coupling and Poisson approximation......Page filename5_0045.djvu
4.13 Exercises. Geometrical probability......Page filename5_0046.djvu
4.14 Solutions to problems......Page filename5_0047.djvu
5.1 Exercises. Generating functions......Page filename5_0056.djvu
5.2 Exercises. Some applications......Page filename5_0057.djvu
5.3 Exercises. Random walk......Page filename5_0058.djvu
5.4 Exercises. Branching processes......Page filename5_0059.djvu
5.6 Exercises. Expectation revisited......Page filename5_0060.djvu
5.7 Exercises. Characteristic functions......Page filename5_0061.djvu
5.8 Exercises. Examples of characteristic functions......Page filename5_0062.djvu
5.9 Exercises. Inversion and continuity theorems......Page filename5_0063.djvu
5.10 Exercises. Two limit theorems......Page filename5_0064.djvu
5.12 Solutions to problems......Page filename5_0065.djvu
6.1 Exercises. Markov processes......Page filename5_0072.djvu
6.2 Exercises. Classification of states......Page filename5_0073.djvu
6.3 Exercises. Classification of chains......Page filename5_0074.djvu
6.4 Exercises. Stationary distributions and the limit theorem......Page filename5_0075.djvu
6.5 Exercises. Reversibility......Page filename5_0076.djvu
6.6 Exercises. Chains with finitely many states......Page filename5_0077.djvu
6.7 Exercises. Branching processes revisited......Page filename5_0078.djvu
6.8 Exercises. Birth processes and the Poisson process......Page filename5_0079.djvu
6.9 Exercises. Continuous-time Markov chains......Page filename5_0080.djvu
6.11 Exercises. Birth-death processes and imbedding......Page filename5_0081.djvu
6.13 Exercises. Spatial Poisson processes......Page filename5_0082.djvu
6.14 Exercises. Markov chain Monte Carlo......Page filename5_0083.djvu
6.15 Solutions to problems......Page filename5_0084.djvu
7.2 Exercises. Modes of convergence......Page filename5_0093.djvu
7.3 Exercises. Some ancillary results......Page filename5_0094.djvu
7.5 Exercises. The strong law......Page filename5_0096.djvu
7.7 Exercises. Martingales......Page filename5_0097.djvu
7.9 Exercises. Prediction and conditional expectation......Page filename5_0098.djvu
7.11 Solutions to problems......Page filename5_0099.djvu
8.3 Exercises. Renewal processes......Page filename5_0105.djvu
8.4 Exercises. Queues......Page filename5_0106.djvu
8.7 Solutions to problems......Page filename5_0107.djvu
9.2 Exercises. Linear prediction......Page filename5_0109.djvu
9.4 Exercises. Stochastic integration and the spectral representation......Page filename5_0110.djvu
9.6 Exercises. Gaussian processes......Page filename5_0111.djvu
9.7 Solutions to problems......Page filename5_0112.djvu
10.2 Exercises. Limit theorems......Page filename5_0115.djvu
10.4 Exercise. Applications......Page filename5_0116.djvu
10.6 Solutions to problems......Page filename5_0117.djvu
11.2 Exercises. M/M/1......Page filename5_0120.djvu
11.5 Exercises. G/G/1......Page filename5_0121.djvu
11.7 Exercises. Networks of queues......Page filename5_0122.djvu
11.8 Solutions to problems......Page filename5_0123.djvu
12.1 Exercises. Introduction......Page filename5_0126.djvu
12.3 Exercises. Crossings and convergence......Page filename5_0127.djvu
12.5 Exercises. Optional stopping......Page filename5_0128.djvu
12.9 Solutions to problems......Page filename5_0129.djvu
13.3 Exercises. Diffusion processes......Page filename5_0134.djvu
13.7 Exercises. Stochastic calculus......Page filename5_0135.djvu
13.8 Exercises. The Itô integral......Page filename5_0136.djvu
13.10 Exercises. Option pricing......Page filename5_0137.djvu
13.12 Solutions to problems......Page filename5_0138.djvu
1.3 Solutions. Probability......Page filename5_0143.djvu
1.4 Solutions. Conditional probability......Page filename5_0145.djvu
1.5 Solutions. Independence......Page filename5_0147.djvu
1.7 Solutions. Worked examples......Page filename5_0148.djvu
1.8 Solutions to problems......Page filename5_0149.djvu
2.1 Solutions. Random variables......Page filename5_0159.djvu
2.4 Solutions. Worked examples......Page filename5_0160.djvu
2.5 Solutions. Random vectors......Page filename5_0161.djvu
2.7 Solutions to problems......Page filename5_0162.djvu
3.2 Solutions. Independence......Page filename5_0166.djvu
3.3 Solutions. Expectation......Page filename5_0169.djvu
3.4 Solutions. Indicators and matching......Page filename5_0170.djvu
3.6 Solutions. Dependence......Page filename5_0173.djvu
3.7 Solutions. Conditional distributions and conditional expectation......Page filename5_0175.djvu
3.8 Solutions. Sums of random variables......Page filename5_0177.djvu
3.9 Solutions. Simple random walk......Page filename5_0178.djvu
3.10 Solutions. Random walk: counting sample paths......Page filename5_0179.djvu
3.11 Solutions to problems......Page filename5_0180.djvu
4.1 Solutions. Probability density functions......Page filename5_0195.djvu
4.2 Solutions. Independence......Page filename5_0196.djvu
4.3 Solutions. Expectation......Page filename5_0197.djvu
4.4 Solutions. Examples of continuous variables......Page filename5_0198.djvu
4.5 Solutions. Dependence......Page filename5_0199.djvu
4.6 Solutions. Conditional distributions and conditional expectation......Page filename5_0201.djvu
4.7 Solutions. Functions of random variables......Page filename5_0203.djvu
4.8 Solutions. Sums of random variables......Page filename5_0207.djvu
4.9 Solutions. Multivariate normal distribution......Page filename5_0209.djvu
4.10 Solutions. Distributions arising from the normal distribution......Page filename5_0210.djvu
4.11 Solutions. Sampling from a distribution......Page filename5_0212.djvu
4.12 Solutions. Coupling and Poisson approximation......Page filename5_0213.djvu
4.13 Solutions. Geometrical probability......Page filename5_0214.djvu
4.14 Solutions to problems......Page filename5_0217.djvu
5.1 Solutions. Generating functions......Page filename5_0238.djvu
5.2 Solutions. Some applications......Page filename5_0240.djvu
5.3 Solutions. Random walk......Page filename5_0242.djvu
5.4 Solutions. Branching processes......Page filename5_0246.djvu
5.5 Solutions. Age-dependent branching processes......Page filename5_0247.djvu
5.7 Solutions. Characteristic functions......Page filename5_0249.djvu
5.8 Solutions. Examples of characteristic functions......Page filename5_0252.djvu
5.9 Solutions. Inversion and continuity theorems......Page filename5_0255.djvu
5.10 Solutions. Two limit theorems......Page filename5_0257.djvu
5.11 Solutions. Large deviations......Page filename5_0261.djvu
5.12 Solutions to problems......Page filename5_0262.djvu
6.1 Solutions. Markov processes......Page filename5_0280.djvu
6.2 Solutions. Classification of states......Page filename5_0283.djvu
6.3 Solutions. Classification of chains......Page filename5_0284.djvu
6.4 Solutions. Stationary distributions and the limit theorem......Page filename5_0289.djvu
6.5 Solutions. Reversibility......Page filename5_0294.djvu
6.6 Solutions. Chains with finitely many states......Page filename5_0295.djvu
6.7 Solutions. Branching processes revisited......Page filename5_0297.djvu
6.8 Solutions. Birth processes and the Poisson process......Page filename5_0298.djvu
6.9 Solutions. Continuous-time Markov chains......Page filename5_0301.djvu
6.11 Solutions. Birth-death processes and imbedding......Page filename5_0305.djvu
6.12 Solutions. Special processes......Page filename5_0307.djvu
6.13 Solutions. Spatial Poisson processes......Page filename5_0309.djvu
6.14 Solutions. Markov chain Monte Carlo......Page filename5_0311.djvu
6.15 Solutions to problems......Page filename5_0312.djvu
7.2 Solutions. Modes of convergence......Page filename5_0331.djvu
7.3 Solutions. Some ancillary results......Page filename5_0334.djvu
7.4 Solution. Laws of large numbers......Page filename5_0338.djvu
7.7 Solutions. Martingales......Page filename5_0339.djvu
7.8 Solutions. Martingale convergence theorem......Page filename5_0340.djvu
7.9 Solutions. Prediction and conditional expectation......Page filename5_0341.djvu
7.10 Solutions. Uniform integrability......Page filename5_0342.djvu
7.11 Solutions to problems......Page filename5_0344.djvu
8.2 Solutions. Stationary processes......Page filename5_0357.djvu
8.3 Solutions. Renewal processes......Page filename5_0358.djvu
8.4 Solutions. Queues......Page filename5_0359.djvu
8.5 Solutions. The Wiener process......Page filename5_0360.djvu
8.7 Solutions to problems......Page filename5_0361.djvu
9.1 Solutions. Introduction......Page filename5_0363.djvu
9.2 Solutions. Linear prediction......Page filename5_0364.djvu
9.3 Solutions. Autocovariances and spectra......Page filename5_0365.djvu
9.5 Solutions. The ergodic theorem......Page filename5_0367.djvu
9.6 Solutions. Gaussian processes......Page filename5_0368.djvu
9.7 Solutions to problems......Page filename5_0369.djvu
10.1 Solutions. The renewal equation......Page filename5_0378.djvu
10.2 Solutions. Limit theorems......Page filename5_0379.djvu
10.3 Solutions. Excess life......Page filename5_0381.djvu
10.5 Solutions. Renewal-reward processes......Page filename5_0383.djvu
10.6 Solutions to problems......Page filename5_0384.djvu
11.2 Solutions. M/M/1......Page filename5_0390.djvu
11.4 Solutions. G/M/1......Page filename5_0392.djvu
11.5 Solutions. G/G/1......Page filename5_0393.djvu
11.7 Solutions. Networks of queues......Page filename5_0394.djvu
11.8 Solutions to problems......Page filename5_0395.djvu
12.1 Solutions. Introduction......Page filename5_0404.djvu
12.3 Solutions. Crossings and convergence......Page filename5_0406.djvu
12.4 Solutions. Stopping times......Page filename5_0407.djvu
12.5 Solutions. Optional stopping......Page filename5_0408.djvu
12.9 Solutions to problems......Page filename5_0411.djvu
13.3 Solutions. Diffusion processes......Page filename5_0419.djvu
13.6 Solutions. Excursions and the Brownian bridge......Page filename5_0421.djvu
13.7 Solutions. Stochastic calculus......Page filename5_0423.djvu
13.8 Solutions. The Itô integral......Page filename5_0424.djvu
13.9 Solutions. Itô\'s formula......Page filename5_0425.djvu
13.10 Solutions. Option pricing......Page filename5_0426.djvu
13.12 Solutions to problems......Page filename5_0428.djvu
Bibliography......Page filename5_0437.djvu
Index......Page filename5_0438.djvu
Back cover......Page p0448.djvu