Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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40 views

Why is the covariance a measure of how much two random variables change together?

In descriptive statistics, the empirical covariance of a point cloud $\left\{(x_i,y_i) : 1\le i\le n \right\}$ is defined as ...
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47 views

Show convergence in distribution of gamma distribution by the central limit theorem

So, I'm not sure how to solve this problem: $X \in \Gamma(a,b)$. Show that $$\frac{X-E[X]}{\sqrt{Var(X)}} \rightarrow^{d} N(0,1)$$ as $a \rightarrow \infty$ Use the central limit theorem. I've come ...
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147 views

Does clever noise exist?

This question is about a random noise, which is called clever if its distribution function satisfies a certain condition. Let $Y_0$ and $Y_1$ be two continuous random variables on the same ...
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145 views

Can we find a correlation between states of a Markov chain?

I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. ...
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22 views

Finding $a$ such that a product of iid $U(0,a)$ converges to $0$ a.s.

Let $a > 0$, let $X_n$, $n \geq 1$, be iid random variables that are uniform on $(0,a)$, and let $Y_n = \prod_{k=1}^n X_k$. Determine, with a proof, all values of $a$ for which $\lim_{n \rightarrow ...
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35 views

Continuity of Mutual Information

Let $(X,Y) \sim P_{X,Y}$ and $(X',Y') \sim Q_{X,Y}$. Suppose $P_{X,Y}$ and $Q_{X,Y}$ both have the same support and we know, \begin{equation*} D(P \Vert Q) < \epsilon \end{equation*} (where ...
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48 views

Removing balls from an urn in pairs: expected number of pairs in which both are red

I am sorry that I cannot make the title more clear. The following is from Sheldon M. Ross: Introduction to Probability Models (11th Edition). I am able to reach the desired answer but actually I don't ...
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27 views

How to show a limit law for record times $N(n) = \min\{j: j > N(n-1), X_j > X_{N(n-1)}\}$, $X_n$ iid.

I have the same question here which I asked a few days ago, but now I have really worked on it and I could not get the solution, so I need some more help. Let $X_1, X_2, \ldots, X_n$ be i.i.d. ...
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52 views

Can the Hamburger moment problem be solved for probability measures?

The hamburger moment problem states that given any real sequence $\{a_n\}$, there exists a positive Borel measure $\mu$ such that $$ a_n =\int_{\mathbb{R}} x^{n}\,d\mu. $$ In other words, the ...
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57 views

Formal examples of Poisson point processes

I am self-studying probability theory and currently the Poisson point process (PPP) gives me hell, firstly because the definition of a point process in general and PPP in particular seems rather ...
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253 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
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29 views

Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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49 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
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49 views

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
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79 views

Definition and derivation of conditional expectation/probability

I read quite a few books introducing the notion of conditional probabilities/expectation by putting a formula out there coming from what they call "intuition". Can someone provide me a good measure ...
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27 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
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22 views

What can be said about a measure with given marginal measures

Let $(X,\mathcal F_X,\mu_X)$, $(Y,\mathcal F_Y,\mu_Y)$ be two measure spaces. Let $\mu$ be a measure on $\bigl(X\times Y, \sigma(\mathcal F_X \times \mathcal F_Y)\bigr)$ such that for each $A \in ...
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106 views

Probability of a sentence when billion monkeys are typing for 10 billion years

Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These ...
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84 views

Clarification of Proof on Kac's Theorem for Characteristic Functions

There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail: Moment generating functions/ Characteristic functions of $X,Y$ ...
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56 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
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60 views

How to select a strongly convergent subsequence from a weak convergent sequence in $L^2$?

Let $(p_n)_{n \in \mathbb N}$ be a sequence of probability density functions, which satisfies i)$$ \partial_t p_n (t,x) = \partial_{xx} (a_n (t,x) p_n (t,x)), $$ where $a_n$ is a sequence upper ...
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86 views

optimization problem gaussian maximizes entropy

Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies, $$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} ...
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53 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
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49 views

Continuity preserved unter expectation? Dominated convergence?

Let $Z$ be a random variable with $0<Z<\infty, 0<\mathbb{E}[Z]<\infty$ and $Z$ be atom-less, i.e. $\mathbb{P}(Z=z)=0$. Further, let $g:\mathbb{R}^+\to\mathbb{R}^+$ be continuous and ...
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89 views

Recovering pmf from characteristic function

I'm having some trouble trying to recover the probability mass function of a discrete random variable from its characteristic function. I have seen that some continuous cases, you can recognize that ...
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25 views

Dolphin in a pool - using Kolmogorov's forward equations

Problem A dolphin swims between 3 different pools, A B and C. The time is spend in each pool, before going to the next one, is Exp(1/2). The possible ways for it to travel is A to B. B to C. C ...
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31 views

Finding $a_n, b_n$ so that a sequence converges in distribution to a nondegenerate random variable.

Now, $X_1, X_2,\dots$ are iid with the same distribution as the chi-squared distribution with one degree of freedom. Find $a_n$ and $b_n$ so that $a_n \left( \max_{1 \leq i \leq n} X_i - b_n \right)$ ...
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59 views

Showing mutual contiguity

The problem: Let $P_n$ and $Q_n$ be the distribution of the mean of a sample of size $n$ from the $N(0, 1)$ and the $N(\theta_n, 1)$ distribution, respectively. Show that $P_n$ and $Q_n$ are ...
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Is this matrix associated with an arbitrary group of events positive semi-definite?

Now I have an arbitrary group of events $X_1,X_2,\ldots,X_m$(with no independence or correlation assumptions, nor distribution knowledge), and define a symmetric matrix $\mathbf{K}$ as below: $$ ...
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53 views

A problem on super/sub martingale

Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time a.s. bounded by $N \lt \infty$. Show that $$ E[|X_T|] \leq 3 \max_{n \leq N} E[|X_n|]$$ I can prove that ...
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34 views

Meaning of (generalized?) differential operator

I am currently reading this paper which makes use of generalized differential operators. As I understood it, the operator $D_x$ works like this: If $F$ is a continuous function on $[a,b]$ and $G$ an ...
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53 views

Bayesian Chain rule

I am going thorugh a Naive Bayes Classifier, and faced the following: $p(y|a,b,c) = \frac{p(a|y,b)*p(y|c)}{p(a|b,c)}$ When I am trying to derive the above, these are my steps: ...
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52 views

Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
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50 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
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109 views

Ito formula for jump proccess

I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have $dS_r=S_{t^-}\mu+S_{t^-}\sigma dB_t +S_{t^-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq ...
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34 views

probability of an event with countably many sample points

The following is Problem 1.9 from A Modern Approach to Probability Theory. Let $A$ be an event which contains countably many sample points. Assume that each of the sample points in $A$ is an event. ...
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51 views

Sample Mean and Sample Variance

Consider a sample of data S obtained by flipping a coin x, where 0 denotes the coin turned up heads, and 1 denotes that it turned up tails. S = {1, 1, 0, 1, 0} What is the sample mean for this data ? ...
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265 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
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40 views

Bounding the size of consecutive sums of independent Bernoullis

Let $\{X_i\}$ be a sequence of independent Bernoulli random variables that take the values 1 and 0 each with probability 1/2. Is the following statement true? For any $\epsilon > 0$, there exists ...
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36 views

Probability of occurrence of two events simultaneously

I have a question with probability of occurrence of two events simultaneously. I have a probability histogram for some events occurring individually. Is it possible to predict the simultaneous ...
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67 views

Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
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49 views

Independence Exercise in Rosenthal

In Rosenthal's, "A First Look At Rigorous Probability Theory", $\exists$ this exercise: Exercise 3.6.19. Let $A_1,\ A_2,\ldots$ be independent events. Let $Y$ be a  random variable which is ...
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68 views

Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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36 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
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56 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
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56 views

Problem regarding Conditional probability

Let $\mathbf{X}$ be an $n-$ dimensional random variable. This variable can be written as $\mathbf{X} = \left[\mathbf{X}_1^T\hspace{5pt}\mathbf{X}_2^T\right]^T$. where, $\mathbf{X}_1$ is $m-$ ...
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78 views

Most important results from pure math in applied probability?

I'm taking a course next semester at my university on applied probability (with relevance to signal and information theory). Although the nature of probability is mostly problem solving and applying ...
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142 views

properties of characteristic function

Let $X,Y$ be two independent random variables having the same distribution, centred and with variance 1, $\phi$ is the characteristic function of $X$ and $Y$. If $X+Y$ and $X-Y$ are independent, show ...
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46 views

Is the plane created by ($\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u})$ continuous?

$f_0$ and $f_1$ are two continuous density functions on $\mathbb{R}$. I wonder if $$(x,y):=\Bigg(\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u}\Bigg)$$ for all $f_0$ and $f_1$ is complete for some ...
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221 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...