Introduction to Probability

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