Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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What is the eigenvalue of stochastic matrix?

I have a stochastic matrix $A \in R^{n \times n}$ whose sum of the entries in each row is 1. When I found out the eigenvalues and eigenvectors for this stochastic matrix, it always happens that one of ...
3
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3answers
977 views

How to obtain the Standard Deviation of a ratio of independent binomial random variables?

X and Y are 2 independent binomial random variables with parameters (n,p) and (m,q) respectively. (trials, probability parameter)
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1answer
363 views

Convergence in mean square from expected value/variance

I'm looking for a proof of the following statement: Given a sequence of independent random variables $X_n$ satisfying $$ \lim_{n\to \infty} E[X_n] = T, $$ where T is a constant, then $$ \lim_{n\to ...
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1answer
202 views

How is the joint distribution of random variables defined and determined?

Suppose there are two random variables $X: \Omega \rightarrow U$ and $Y: \Omega \rightarrow V$ with probability space $(\Omega, \mathcal{F}, P)$, measurable spaces $(U, \mathcal{F}_u)$ and $(V, ...
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1answer
158 views

Comparison between Rademacher average and random average

Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of ...
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5answers
469 views

When can a random variable be written as a sum of two iid random variables?

Suppose $X$ is a random variable; when do there exist two random variables $X',X''$, independent and identically distributed, such that $$X = X' + X''$$ My natural impulse ...
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1answer
96 views

Reasoning error in “maximin” type problem

I have a problem that I have approached two different ways. The approaches give me two different answers. I have matched the answers against a simulation, so I think I know which one is right. But, ...
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1answer
97 views

Rate of change of $\mathbb{E}X_{\frac{2^n}{2},2^n}$ as $n$ increases?

I am trying to get an equation that will show the rate of change of the expected value of $\frac{2^n}{2}$th lowest of $2^n$ draws from $X$ as $n$ increases (where $n >1$). Let's call the order ...
3
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1answer
469 views

Independence and Conditional Expectations for three random variables, (How) does $X$ independent from $(Y,Z)$ imply $E(X/Y | Z)= E(X) E(1/Y|Z)$?

I have three random variables. Let $X$ be independent from $(Y,Z)$ ($:=\sigma(Y,Z))$, $Y\neq0$ a.s. Q: How can I proof (if it holds), that $E\left(\frac{X}{Y} \mid Z\right)=E(X) ...
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1answer
206 views

Strange expression for kind of independence

In a paper, I find the expression "Let $$\{X(Y_k) | Y_k\}_{k=0,1, \cdots}$$ be mutually independent" Q: What does this notation mean? Does somebody know it? Is it some kind of conditional ...
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1answer
147 views

Is there a simple argument for why a random symmetric matrix has distinct eigenvalues?

Lets generate a random symmetric matrix $A$ by generating the entries of a random matrix $Z$ iid from some continuous distribution, and setting $A=(1/2)(Z+Z^T)$. I think its true that $A$ should have ...
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1answer
81 views

Feller continuity of the stochastic kernel: compact set

This question is an extension of Feller continuity of the stochastic kernel. Nate Eldredge provided a nice counterexample, but I failed trying to extend it to the compact set $B$. The setting is the ...
4
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1answer
294 views

Feller continuity of the stochastic kernel

Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
3
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1answer
149 views

Bayesian analysis of the Venice Doge elections

Does anyone know of a Bayesian (or a classical) analysis of the Venetian Doge election system? I am looking mainly for chances of subversion, chances for a candidate to be elected at each stage, or ...
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1answer
715 views

How far can probability intransitivity be stretched?

Once upon a time I read about nontransitive dice - sets of dice where "is more likely to roll a higher number than" is not a transitive relation. After the surprise wore off, I wondered - just how far ...
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0answers
144 views

Family of measure that admits a continuous density

This question is a generalization of an example provided in Absolute continuous family of measures. Consider a metric space $(X,\rho)$ with a Borel $\sigma$-algebra $\mathcal B(X)$. Consider a ...
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1answer
199 views

Convergence of inner product using Cauchy-Schwarz

I'm reading a paper in which the following argument is made (in the proof of Theorem 7). I will try to provide just the essentials necessary to ask my question, in particular omitting the ...
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1answer
132 views

Absolute continuous family of measures

Consider the following family of measures on $(\mathbb R,\mathcal B(\mathbb R))$: $$ K_x(A) = \begin{cases} \int\limits_A \frac{1}{|x|\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy&,\text{ if }x\neq 0, ...
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1answer
151 views

Can the following probabilistic argument about eigenvalues be made rigorous?

Consider the following $n \times n$ matrix $$ \left( \begin{matrix} 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 1/2 & 0 & 1/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 ...
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1answer
171 views

$\sqrt{ 1 - \zeta } = \sqrt{ \zeta }$?

In this paper (page 6) I'm reading, the author has a uniform random variable $\zeta$ which takes on values between 0 and 1. He computes $$ 2 \arccos( \sqrt{ 1 - \zeta } ) $$ But isn't that the ...
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0answers
87 views

Reachability for Markov process, continuous time

Let $X$ be a strong Markov process in the continuous time with a state space $\mathbb R^n$. Consider a reachability problem for this process, i.e. $$ v(x):=\mathsf P_x\{X_t\in A\text{ for some }0\leq ...
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2answers
981 views

Tough probability question: Fair and Unfair die rolling

Have been trying for the last three hours, and going nuts. Please provide HINTS only, not the solution (or answer). I'm doing this by a dumb approach, way too much of calculations and excel madness. ...
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3answers
133 views

Independence Details for 3 random variables

Suppose I have three random variables X,Y,Z. X is independent of Y and of Z. I want to conclude, that $X$ independent from $(Y,Z)$. $(Y,Z):= \sigma(Y,Z)=\sigma( \{ Y^{-1}(A) | A\in \mathcal{E}_1 \} ...
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2answers
248 views

Detail in Conditional expectation on more than one random variable

I have $E(X|Y,Z)=0$, $X$ independent of $Y$ and of $Z$ and I want to conclude that $E(X)=0$ ($X,Y,Z$ are real-valued random variables). Okay it seems quite obvious, but if I try to make a strict ...
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0answers
225 views

Convergence of moment-generating function of weighted sums of random variables

This is a continuation of my earlier question. Once again, let $c_n$ be a sequence of positive real numbers such that $$\sum^{\infty}_{n=1}{c_n} = \infty, \qquad \sum^{\infty}_{n=1}{c_n^2} < ...
2
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1answer
974 views

Rule with independent random variables and conditional expectations

I want to use a rule for conditional expectation I found in (German) wikipedia, not in my script/textbook of probability theory, I guess it should be simple and follow more or less straight from the ...
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2answers
860 views

Calculate the probability of a simple event

I'm beginning to study probability and an exercise in the study guide that asks me to calculate: What is the probability that the month January, of one year randomly selected have only four Sundays? ...
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2answers
433 views

Convergence of variance and mean of weighted sums of random variables

Let $c_n$ be a sequence of positive real numbers such that $$\sum^{\infty}_{n=1}{c_n} = \infty, \qquad \sum^{\infty}_{n=1}{c_n^2} < \infty.$$ Let $X_n$ be a family of i.i.d random variables with ...
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3answers
891 views

Is this a Delta Function? (and Delta as limit of Gaussian?)

I have a set of users that generate calls. If I assign the same probability to each user, they have identical call generation probability which can be defined as $\delta$. These callers are chosen ...
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1answer
199 views

How to understand conditional expectation on “current iterate” (as conditioned on a random variable? or on certain sets?)

I have a sequence of random variables and want to work out the details of a proof. The author is not so accurate, and he uses conditional expectation values $E( Y | X_k)$ where $X_k$ is the current ...
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1answer
193 views

Finiteness of conditional expectation if expectation is finite

I have $E(X) < \infty$. Under which conditions follows that $E(X|A)<\infty$ ? (A is an event of the form {$Y=y$} if it should matter) If I can use the formula $E(X|A)=\frac{E(X 1_A)}{P(A)}$ ...
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1answer
2k views

Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
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1answer
261 views

How can I prove this expression not to be a characteristic function

Let $\phi$ be a function of two real arguments defined as follows: $$ \phi\left(t_1, t_2\right) = \exp \left(-t_1^2-t_2^2 +i \frac{t_1}{3}\frac{ t_1^2-3 t_2^2 }{t_1^2+t_2^2} \right)$$ and whenever ...
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3answers
208 views

Elementary Probability Question: Conditional case, with AT LEAST clause

Q. An anti aircraft gun fires at a moving enemy plane with 4 successive shots. Probabilities of the shots S1, S2, S3 and S4 hitting the plane are 0.4, 0.3, 0.2, 0.1 respectively. (a) What is the ...
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2answers
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Distribution of Max(X_i) | Min(X_i), X_i are iid uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) | \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$ I agree that the probability ...
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0answers
265 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
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4answers
2k views

Can a Dirac delta function be a probability density function of a random variable?

Can the Dirac delta function (or distribution) be a probability density function of a random variable. To my knowledge, it seem to satisfy the conditions. To my interpretation getting a positive real ...
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0answers
104 views

If $\frac{X_n}{b_n} \to 0$ almost surely where $0 < b_n \uparrow \infty$ then $\frac{\max_{1 \le j \le n}|X_j|}{b_n} \to 0$ almost surely?

Let $X_1, X_2, ...$ be a sequence of random variables (finite almost surely), and let $0 < b_n \uparrow \infty$. As part of solving a problem I seem to have shown that $\frac{X_n}{b_n} \to 0$ ...
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1answer
306 views

Chebyshev Inequality and Inequality Question

I have a question about the Chebyshev inequality in probability. Specifically, I am concerned with the term inside the probability function. I agree with the following: Let $Y = (X-EX)^2$. Then Y is ...
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1answer
135 views

How to measure the amount of uncertainty

What are the possible measures of uncertainty for a discrete variable X=(x1, x2, ... xn), where values are defined by the alphabet - xi ∈ A, given probabilities p(xi) = P(X = xi) change over time? ...
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2answers
109 views

convergence of random variables

This is related to this question. How exactly does one go about computing the limits in the answer to the linked question. Thanks. P.S: I would have commented on the linked question, but I don't ...
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1answer
87 views

An example of a projective sequence of measures on non-Borel spaces that does not extend to a probability on the product space?

Daniell proved a theorem on the existence of random sequences (see page 13 of these notes): Let $(S_n,\mathbf{S_n})$ be a sequence of Borel spaces and let $\mu_n$ be a projective sequence of ...
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2answers
188 views

Marginal Distribution

Let X be a normal random variable with mean $\mu$ and variance $\sigma^2$. Let us further assume that $\mu \sim \mathcal N(\mu_p, \sigma_p^2)$ and its prior distribution is $\pi(\mu)$. The variables ...
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4answers
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A good book on Statistical Inference?

Anyone can suggest me one or more good books on Statistical Inference (estimators, UMVU estimators, hypotesis testing, UMP test, interval estimators, ANOVA one-way and two-way...) based on rigorous ...
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1answer
121 views

Clarification of “martingales null at zero”

Can someone please help me to clarify a definition. I am reading about stochastic process from a book and I come across the statement: "martingales null at zero". What does this mean? Please be a ...
4
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2answers
798 views

Can I normalize KL-divergence to be $\leq 1$?

The Kullback-Leibler divergence has a strong relationship with mutual information, and mutual information has a number of normalized variants. Is there some similar, entropy-like value that I can use ...
3
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1answer
322 views

Using Dominated Convergence Theorem

I'm trying to follow a proof of Lévy's Continuity Theorem. Let $X_n$ be a sequence of random variables with characteristic functions $\phi_n$, such that $\phi_n\to\phi$ pointwise and such that $\phi$ ...
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2answers
303 views

pdf of power of rayleigh variables

I'm trying to compute the pdf of the power of a rayleigh distributed random variable. So, let $X$ be distributed as Rayleigh $$ X \sim \frac{x}{\sigma^2} e^{-\frac{x^2}{2\sigma^2}}, x\geq 0 $$ let ...
3
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1answer
284 views

Something connected with Ulam's tightness theorem

Well known theorem of Ulam says, that each probability measure $\mu$ defined on Borel subsets of polish space $X$ satisfies the following condition: for each $\epsilon>0$ there is a compact subset ...
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2answers
139 views

Adding two Probabilities

Suppose i have Two Text Articles (article 'a' => x words, article 'b' => y words) i find the total number of words in 'a' = x the total number of occurances of the word "the" in the article = x1 ...