3
votes
2answers
58 views

What is the best book to learn statistics?

Right now I'm taking a 3 part course on probability and statistics using Schverish & Degroot Probability and Statistics and it is just not helpful. For the first part, which was on Probability, I ...
1
vote
1answer
28 views

References for a second course in probability theory

I need a probability book that treats all the arguments from the point of view of the measure theory and the Lebesgue integral. I've the basis of "naive" probability theory and of measure theory so I ...
6
votes
6answers
343 views

Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
0
votes
0answers
23 views

Learning resources for Probability Distributions/Models

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
5
votes
3answers
72 views

A list of different measures of distance/difference/dissimilarities/similarity of two probability distributions?

I wanted to know more about the different methods for comparing the similarities of two probability distributions P and Q. I wanted a list of the different methods that exist for comparing ...
11
votes
1answer
157 views

Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
1
vote
0answers
31 views

Introductory Statistics/Probability Reference Request

I am a PhD student in mathematics with a background in Pure Mathematics. As such I have pretty much zero background in statistics or anything other than the most basic probability (i.e., what one ...
0
votes
0answers
16 views

Estimating the magnitude of a change in a non-stationary stochastic process

In this paper by Adams and MacKay, they present an algorithm for the online detection of change-points in a stochastic process subject to some hypotheses. Their algorithm gives both the predictive ...
2
votes
0answers
34 views

Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let $$ h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}| $$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = ...
1
vote
2answers
43 views

References for probability using Calculus

I have to teach a Calculus class (details on the syllabus below) and I want to add some applications to other sciences. But I would like to avoid the classical physics examples, because the physics ...
0
votes
0answers
21 views

Is it possible to group linear maps by similarity or likelihood that they are identical?

Given a set of linear maps $f: x \rightarrow y$, is there a way to (statistically or otherwise) determine how likely two linear maps are identical even if I do not have enough data to determine $f$? ...
1
vote
1answer
23 views

Guidance regarding probability

Does anyone know any good problems collection for elementary probability to be used along with Stirzaker ? I currently have Fifty Challenging Problems in Probability by Mosteller, but that is not ...
4
votes
5answers
138 views

Please recommend a nice and concise math book on probability theory.

My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in ...
0
votes
2answers
63 views

Statistics Major

I need multiple books on statistics, not just one book, so I'm looking for a big course on statistics, can anybody help me with books, book list or anything like that? I want to study it as if it's ...
1
vote
1answer
70 views

Statistic Textbooks

What is a good textbook for introductions to continuous and discrete distributions? The one that my university offers is a thin scrap put together by the department. Could I get some recommendations ...
2
votes
0answers
38 views

Product Involving Sines

I'm studying the following product: $$p(a,\omega)=\prod_{k=1}^{\infty}a\sin (k\omega\pi),\quad \omega \in \Bbb R,\quad a\in \Bbb R_+.$$ It's easy to see that for $a\in (0,1]$ this product diverges to ...
1
vote
0answers
28 views

book related query

I have been solving a lot of problems in algebra, calculus, probability and statistics. But have never encountered problems that consist of every mathematical field mentioned above (at max two ...
8
votes
2answers
163 views

An extrasensory perception strategy :-)

Inspired by classical Joseph Banks Rhine experiments demonstrating an extrasensory perception (see, for instance, the beginning of the respective chapter of Jeffrey Mishlove book “The Roots of ...
8
votes
3answers
151 views

In search of Probability text recommendations

The probability class I recently finished (taught at an upper-undergraduate or lower-graduate level) used the text by Grimmett and Stirzaker. I really disliked this book. I am familiar with measure ...
2
votes
1answer
35 views

what is a list of probability puzzle books that focus solely on probability?

Referring back to my problem: Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem? What is a list of probability puzzle books that ...
1
vote
2answers
131 views

Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem?

For recreational purpose, i haven't seen a interesting elemetary probability question quite a while. Is there any surprising elementary probability problem that result in surprising solution like the ...
3
votes
4answers
125 views

Self study on probability and statitistics

i know there are similar questions already, but i specifically need a book that covers these topics: Combinatorics, conditional probability, Bayes theorem, random variables, joint probability, ...
0
votes
0answers
23 views

Reference request for Markov chains [duplicate]

I'm reading about Markov chains in Grimmett/Stirzaker, and I find the notation in that book confusing and the exposition only decent. Does anyone know a better reference?
4
votes
0answers
33 views

How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
1
vote
1answer
66 views

Book for probability and various probability distribution functions.

Please suggest a book/books where i can understand Probability theory (with lots of example and solution) examples on permutations and combinations. list of all probability distribution functions, ...
1
vote
1answer
30 views

Necessary and sufficient condition for $G(z)$ to be a probabilty generating function.

Given a function $G(z)$, what are the sufficient and necessary conditions for being a $G(z)$ to be a probability generating function. Few necessary condition which I know of are $G(1) = 1$ All the ...
2
votes
1answer
100 views

Complex Analysis and Probability Theory

My question is a general one. I know that in complex analysis we find some very powerful theorems but given that my main area of study is Statistics and Probability, does complex analysis have ...
3
votes
2answers
69 views

$\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$

A question in my probability class: Let $X$ have the binomial distribution $\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$. Show that $X$ is uniformly distributed on $\{0,1,\dotsc, n\}$. ...
0
votes
0answers
28 views

Bootstrap sampling (i.e. sample N with replacement) - distribution of histogram

In bootstrap sampling, we have $N$ items and we perform random sampling with replacement $N$ times. The resulting sample could be summarised by a histogram illustrating the number of items which were ...
2
votes
2answers
145 views

Looking for a book on Probability and Statistics.

I am looking for a book or website on mathematical theory of probability and statistics for preparation of an examination. The syllabus written in the unit 4 of this document. Only multiple choice ...
0
votes
2answers
199 views

Good introductory probability book for graduate level?

Would you please suggest a good, readable introductory probability book for graduate level ? I have Shiryaev 's Probability with me, however i want to find another one. Preferably with solution manual ...
1
vote
0answers
41 views

Convergence of EM algorithm with continuous hidden variables

I am interested in a proof of convergence of EM algorithm when hidden variables are continuous. I found a proof for a case of discrete hidden variables, but I cannot find it for continuous case. Do ...
2
votes
1answer
1k views

Probability of finding at least k consecutive heads in N coin tosses?

There are quite a few topics on this question already but I couldn't find a well-explained solution. Please point me towards some relevant literature or theory to analyze this problem. $K$ ...
1
vote
0answers
53 views

When is $\mbox{Var}(X|\mathbf Y = \mathbf y) < \mbox{Var}(X)$ for all $\mathbf y$?

Consider a random variable $X$ and random vector $\mathbf Y$. The law of iterated variance states that $$ \mbox{Var}(X) = E\left\{\mbox{Var}(X|\mathbf Y)\right\} + \mbox{Var}\left\{E(X|\mathbf ...
4
votes
1answer
117 views

Suggestions for learning probability and statistics

The Short question: Where can I find a book for probability and statistics book that teaches them from scratch in a rigorous (very important condition) way ? The book must not be elementary, but it ...
2
votes
1answer
45 views

Reference request: the Gaussian is determined by its moments

It came up in a paper I am coauthoring that a Gaussian distribution is determined by its moments (and in particular that a probability distribution we are considering converges weakly to the ...
1
vote
1answer
150 views

Looking for first course textbooks on probability and statistics for math majors

I am taking a probability and statistics course soon and would like to find a text book that is targeted more towards math majors rather than engineers (which is what this class is). The book my ...
6
votes
2answers
185 views

Probability measure on subset of natural numbers…

How one would define a probability measure on all subsets of natural numbers, which is finite-additive and such that the variables $\chi_p(n)=\left\{\begin{matrix} 1 & p|n \\ 0 & \text{esle} ...
1
vote
0answers
150 views

References for problems related to uniformly distributed points/ arcs on circle

I am looking for references to problems related to computing the statistical properties of uniformly randomly chosen points or arcs on the unit circle, e.g.: On a unit circle, $n$ points are ...
2
votes
0answers
195 views

The most fundamental papers in stochastic analysis

I have soft a question. What papers will be good to on start and allow me to make little step into research, without harm for reader. I am interested in an stochastic analysis. I am looking for ...
2
votes
2answers
259 views

Expected state of a markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
1
vote
1answer
68 views

Reference for the Law of the Unconscious Statistician?

Does anyone know of a reference (a book or journal article) for the Law of the Unconscious Statistician?
0
votes
0answers
34 views

convergence to types theorem - reference

I'm looking for a good reference to the following (or similar) statement of the convergence to types theorem (good meaning a book or a research paper with a proof of this theorem) Let $X_n$ be a ...
3
votes
1answer
117 views

Quickest path to proof of Central Limit Theorem?

I'm looking for a quick approach to a rigorous proof of the central limit theorem of probability theory. (I imagine that this would have to be a paper or monograph with this as its main goal, ...
2
votes
1answer
29 views

Question about convergence in probability (topic confusion)

I'm taking second year stats and was introduced the below concepts For the third one, we use that to estimate the mean squared error in the case where the estimator is a nonlinear function of the ...
1
vote
0answers
42 views

Conditions for the ground state of Gibbs ensemble not to be “degenerate”

I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions. I am looking for books or other resources ...
3
votes
0answers
65 views

Some long and good prerequisite textbooks to the graduate probability textbook by shiryaev and boas?

Some long and good prerequisite textbooks to the graduate probability textbook by shiryaev and boas? It seems that it have a big gap between this graduate textbooks and the easier ones.
1
vote
0answers
29 views

literature to learn more on ergodic harris recurrent chains with an atom

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's ...
3
votes
2answers
131 views

uniform integrability of a squared sum of iid variables

I'm trying to prove that if $X_i$ are independent, identically distributed random variables such that $E X_i = 0$ and $E X_i^2 < \infty$ then the sequence $\frac{(\sum_{i=1}^{n} X_i)^2}{n}$ is ...
1
vote
0answers
52 views

Weak Law of Large Numbers proof

I want to know if there is a proof of the Weak Law of Large Numbers without using the Chebyshev's Inequality? please can anyone give me some references