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

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