Course material for Richard Weber’s course on Probability for first year mathematicians at Cambridge. You can also check Richard’s blog (a former colleague of my dad) here. It also includes exams question. This is the base material that needs to be mastered before being accepted in the prestigious Tripo III curriculum at Cambridge. The book (PDF) can be downloaded here. Below is the table other contents. Other similar books can be found here.
Source: see section 24.3 in the book
1 Classical probability
1.1 Diverse notions of `probability’
1.2 Classical probability
1.3 Sample space and events
1.4 Equalizations in random walk
2 Combinatorial analysis
2.1 Counting
2.2 Sampling with or without replacement
2.2.0.1 Remarks.
2.3 Sampling with or without regard to ordering
2.4 Four cases of enumerative combinatorics
3 Stirling’s formula
3.1 Multinomial coefficient
3.2 Stirling’s formula
3.3 Improved Stirling’s formula
4 Axiomatic approach
4.1 Axioms of probability
4.2 Boole’s inequality
4.3 Inclusion-exclusion formula
5 Independence
5.1 Bonferroni’s inequalities
5.2 Independence of two events
5.2.0.1 Independent experiments.
5.3 Independence of multiple events
5.4 Important distributions
5.5 Poisson approximation to the binomial
6 Conditional probability
6.1 Conditional probability
6.2 Properties of conditional probability
6.3 Law of total probability
6.4 Bayes’ formula
6.5 Simpson’s paradox
6.5.0.1 Remark.
7 Discrete random variables
7.1 Continuity of $P$
7.2 Discrete random variables
7.3 Expectation
7.4 Function of a random variable
7.5 Properties of expectation
8 Further functions of random variables
8.1 Expectation of sum is sum of expectations
8.2 Variance
8.2.0.1 Binomial.
8.2.0.2 Poisson.
8.2.0.3 Geometric.
8.3 Indicator random variables
8.4 Reproof of inclusion-exclusion formula
8.5 Zipf’s law
9 Independent random variables
9.1 Independent random variables
9.2 Variance of a sum
9.3 Efron’s dice
9.4 Cycle lengths in a random permutation
9.4.0.1 Names in boxes problem.
10 Inequalities
10.1 Jensen’s inequality
10.2 AM–GM inequality
10.3 Cauchy-Schwarz inequality
10.4 Covariance and correlation
10.5 Information entropy
11 Weak law of large numbers
11.1 Markov inequality
11.2 Chebyshev inequality
11.3 Weak law of large numbers
11.3.0.1 Remark.
11.3.0.2 Strong law of large numbers
11.4 Probabilistic proof of Weierstrass approximation theorem
11.5 Probabilistic proof of Weierstrass approximation theorem
11.6 Benford’s law
12 Probability generating functions
12.1 Probability generating function
12.2 Combinatorial applications
12.2.0.1 Dyck words.
13 Conditional expectation
13.1 Conditional distribution and expectation
13.2 Properties of conditional expectation
13.3 Sums with a random number of terms
13.4 Aggregate loss distribution and VaR
13.5 Conditional entropy
14 Branching processes
14.1 Branching processes
14.2 Generating function of a branching process
14.3 Probability of extinction
15 Random walk and gambler’s ruin
15.1 Random walks
15.2 Gambler’s ruin
15.3 Duration of the game
15.4 Use of generating functions in random walk
16 Continuous random variables
16.1 Continuous random variables
16.1.0.1 Remark.
16.2 Uniform distribution
16.3 Exponential distribution
16.4 Hazard rate
16.5 Relationships among probability distributions
17 Functions of a continuous random variable
17.1 Distribution of a function of a random variable
17.1.0.1 Remarks.
17.2 Expectation
17.3 Stochastic ordering of random variables
17.4 Variance
17.5 Inspection paradox
18 Jointly distributed random variables
18.1 Jointly distributed random variables
18.2 Independence of continuous random variables
18.3 Geometric probability
18.4 Bertrand’s paradox
18.5 Last arrivals problem
19 Normal distribution
19.1 Normal distribution
19.2 Calculations with the normal distribution
19.3 Mode, median and sample mean
19.4 Distribution of order statistics
19.5 Stochastic bin packing
20 Transformations of random variables
20.1 Convolution
20.2 Cauchy distribution
21 Moment generating functions
21.1 What happens if the mapping is not 1–1?
21.2 Minimum of exponentials is exponential
21.3 Moment generating functions
21.4 Gamma distribution
21.5 Beta distribution
22 Multivariate normal distribution
22.1 Moment generating function of normal distribution
22.2 Functions of normal random variables
22.3 Multivariate normal distribution
22.4 Bivariate normal
22.5 Multivariate moment generating function
22.6 Buffon’s needle
23 Central limit theorem
23.1 Central limit theorem
23.1.0.1 Remarks.
23.2 Normal approximation to the binomial
23.3 Estimating $pi$ with Buffon’s needle
24 Continuing studies in probability
24.1 Large deviations
24.2 Chernoff bound
24.3 Random matrices
24.4 Concluding remarks
Appendices
A Problem solving strategies
B Fast Fourier transform and p.g.fs
C The Jacobian
D Beta distribution
E Kelly criterion
F Ballot theorem
G Allais paradox
H IB courses in applicable mathematics
Credit: Data Science Central By: Capri Granville
