0
votes
0answers
28 views

On achieving the maximal correlation

I am reading the famous paper of Renyi, entitled "On measures of dependence" (see here1). He redefined the maximal correlation in a very general form for both discrete and continuous random ...
2
votes
1answer
45 views

Where can I find a set of probability problems?

Is there a database of solved probability problems available? I am currently studying probability (and statistics) and, while I think I have a decent grasp of permutations, combinations, conditional ...
0
votes
0answers
14 views

Looking for specific book/thesis about Ising models

I am looking for a book/thesis I read about Ising models. I'm pretty sure it was done in the 1970 or maybe 1980s, since it was definitely pre-LaTeX (it looked type-written). The book also included ...
4
votes
1answer
35 views

Cramer's theorem reference request

I'm looking for a proof of Cramer's theorem that states the following: Let $X,Y$ two independent random variables such that $X+Y$ is normal distributed, then $X$ and $Y$ are normal distributed. ...
4
votes
2answers
69 views

Probability that two random permutations of an $n$-set commute?

From this MathOverflow question: It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. -- Benjamin ...
0
votes
1answer
18 views

Good references on the convergence of Markov chains with countably infinite state space

I'm looking for good references on the convergence time of a Markov chain with a countably infinite state space. Most books deal with finite state Markov chains, whose theory is well established. ...
1
vote
0answers
21 views

Standard deviation and related quantities

By definition, standard deviation is the square root of the variance. There is some common terminology for the quantities $$\mathbb{E}(|X-\mathbb{E}X|^p)^{1/p} $$ for $p \geq 1$? Or, they are just ...
2
votes
1answer
142 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
2
votes
1answer
42 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
0
votes
0answers
40 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
2
votes
2answers
49 views

Bayesian learning for input “If A, then B.”

Can anyone point me to literature on Bayesian learning when the new information has the form “If A, then B”? I’m familiar with the rule that after one learns X, posterior probability P(Y) equals prior ...
0
votes
0answers
12 views

deterministic limit of gaussian distribution

Let $a$ be a random variable over some set $A$, and let $\mathcal A \subseteq A$ be an event. Let $\mathcal E \subset \mathbb R^n$ be another event, and let $x_1, \dots, x_n$ be several Gaussian ...
3
votes
2answers
148 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
45 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 ...
7
votes
6answers
425 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
29 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
111 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 ...
16
votes
1answer
7k 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
50 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
19 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
39 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
60 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
26 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 ...
5
votes
5answers
181 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
92 views

Statistics Major [closed]

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
82 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
39 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
29 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 ...
9
votes
2answers
184 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
194 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
158 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
224 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
25 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
43 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
82 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
33 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
142 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
38 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
211 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
260 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
46 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
2k 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
55 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
157 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
47 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
197 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
209 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} ...