CAMBRIDGE MATHEMATICAL TEXTBOOKS
Contents
Preface
To the instructor
From gambling to an essential ingredient of modern science and society
Experiments with random outcomes
1.1. Sample spaces and probabilities
1.2. Random sampling
1.3. Infinitely many outcomes
1.4. Consequences of the rules of probability
1.5. Random variables: a first look
1.6. Finer points 4>
Exercises
Further exercises
Challenging problems
2
Conditional probability and independence
2.1. Conditional probability
2.2. Bayes' formula
2.3. Independence
A В
2.4. Independent trials
r(N > 7) = = И = £ (I)"1 i = HI)7 E (If = frr = (I)7 -
2.5. Further topics on sampling and independence
2.6. Finer points 4»
Exercises
Further exercises
w=e=A©1-
Challenging problems
Random variables
3.1. Probability distributions of random variables
fix) =
I0’
3.2. Cumulative distribution function
3.3. Expectation
E^"1 = E^Uk)=^ (У?)
=£да=ч.
3.4. Variance
Summary of properties of random variables
3.5. Gaussian distribution
Fact 3.56.
Density function
Range, support, and possible values of a random variable
Continuity of a random variable
Properties of the cumulative distribution function
Expectation
Moments
Quantiles
Exercises
Section 3.1
Section 3.4
Section 3.5
Further exercises
Challenging problems
Approximations of the binomial distribution
4.1. Normal approximation
The CLT for the binomial and examples
/ e 2 dx. a 'flu
Continuity correction
A partial proof of the CLT for the binomial ♦
4.2. Law of large numbers
4.3. Applications of the normal approximation
Confidence intervals
Polling
4.4. Poisson approximation
Poisson approximation of counts of rare events
Comparison of the normal and Poisson approximations of the binomial
4.5. Exponential distribution
Derivation of the exponential distribution ♦
4.6. Poisson process ♦
( rwe =
Error bound in the normal approximation
Weak versus strong law of large numbers
The memoryless property
Spatial Poisson processes
Exercises
Section 4. 1
Section 4. 2
Section 4. 3
Section 4. 4
Section 4. 5
Section 4.6
Further exercises
Challenging problems
Transforms and transformations
5.1. Moment generating function
Calculation of moments with the moment generating function
Equality in distribution
w)]=E =E ^x]p[Y=x]=л].
Identification of distributions with moment generating functions
5.2. Distribution of a function of a random variable
Discrete case
Continuous case
Generating random variables from a uniform distribution ♦
Equality in distribution
Transforms of random variables
Section 5.1
Section 5.2
Further exercises
Challenging problems
Joint distribution of random variables
6.1. Joint distribution of discrete random variables
6.2. Jointly continuous random variables
J—oo J—oo Jo \Jo / Jo
• • • / f(xb... ,Xj_bx, Xj+1,..., xn) dXi... dXj-1 dxj+1 ...dxn.
fx(x]= [ fx,y(x,y]dy. (6.13)
J—OQ
dx.
Uniform distribution in higher dimensions
Nonexistence of joint density function ♦
6.3. Joint distributions and independence
Further examples: the discrete case
Further examples: the jointly continuous case
6.4. Further multivariate topics ♦
Joint cumulative distribution function
J—OQ J —OO
Standard bivariate normal distribution
Infinitesimal method
Transformation of a joint density function
fR(r) =
Zero probability events for jointly continuous random variables
Measurable subsets of Rn
Joint moment generating function
Section 6.1
Section 6.2
Section 6. 3
Section 6. 4
Further exercises
Challenging problems
Sums and symmetry
7.1. Sums of independent random variables
7.2. Exchangeable random variables
ад еВ1,Х2еВ2,...,^,еВя)
Independent identically distributed random variables
Sampling without replacement
7.3. Poisson process revisited ♦
Exercises
Section 7. 1
Section 7. 2
Section 7. 3
Further exercises
Challenging problems
8
Expectation and variance in the multivariate setting
8.1. Linearity of expectation
8.2. Expectation and independence
Sample mean and sample variance
Coupon collector's problem
8.3. Sums and moment generating functions
8.4. Covariance and correlation
Variance of a sum
= 53 ад - Mx,)2]+52 52 - ^)(Xj - MX, )]
Uncorrelated versus independent random variables
Properties of the covariance
Correlation
8.5. The bivariate normal distribution ♦
Independence and expectations of products
Multivariate normal distribution
Details of expectations and measure theory
Exercises
Section 8. 1
Section 8. 2
Section 8. 3
Section 8. 4
Section 8. 5
Further exercises
Challenging problems
Tail bounds and limit theorems
9.1. Estimating tail probabilities
9.2. Law of large numbers
£ZW
9.3. Central limit theorem
Proof of the central limit theorem ♦
9.4. Monte Carlo method ♦
Confidence intervals
J a JW
Generalizations of Markov's inequality
The strong law of large numbers
Proof of the central limit theorem
Error bound in the central limit theorem
Section 9.1
Section 9. 2
Section 9. 3
Section 9. 4
Further exercises
Challenging problems
lim E
10
Conditional distribution
10.1. Conditional distribution of a discrete random variable
Conditioning on an event
Conditioning on a random variable
( n nn~m~e
Constructing joint probability mass functions
Marking Poisson arrivals
e~P^(p^ е-Р^Ык2 е~Р^(р3к)кз fej Ы
10.2. Conditional distribution for jointly continuous random variables
Constructing joint density functions
Justification for the formula of the conditional density function ♦
10.3. Conditional expectation
Conditional expectation as a random variable
J—OQ
J—OQ
Multivariate conditional distributions
Examples
Conditioning and independence
Conditioning on the random variable itself
£(X|X) = X.
Conditioning on a random variable fixes its value ♦
=E
10.4. Further conditioning topics ♦
Conditional moment generating function
Mixing discrete and continuous random variables
Conditional expectation as the best predictor
ВДП
Random sums
Trials with unknown success probability
J a
Conditioning on multiple random variables: a prelude to stochastic processes
P($n = Xn, Yn+1 = Xn+1 Xn) P(Sn = Xn) P(Yn+l = -fn+1 %n) P(Sn = xn) P(Sn = xn)
Conditional expectation
Exercises
Section 10.1
Section 10.2
Section 10.3
Section 10.4
Further exercises
Challenging problems
Appendix A
Things to know from calculus
Appendix В
Set notation and operations
Exercises
Challenging problems
Appendix С
Counting
Fact C.5. Let Ab A2,... ,A„ be finite sets.
# (Ai x A2 x • • • x An) = (#Ai) • (#A2) • • • (#A„) = f[(#Ai).
Practical advice
Proof by induction
Exercises
Exercises to practice induction
Appendix D
Sums, products and series
Sum and product notation
Infinite series
Changing order of summation
(ii)if 5222lflijl
= 2222^-
Summation identities
Exercises
^EE
(c) EE(7 + 2fe + £)
Challenging problems
Appendix E
Table of values for Ф(х)
Appendix F
Table of common probability distributions
Answers to selected exercises
Chapter 1
Chapter 2
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Airily) =
Appendix В
Appendix C
Bibliography
Index