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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1answer
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How to compute the expectation $\mathbb{E}[X^2]$

Consider a set $\Omega$ of all permutations on the set $\{1,2,\cdots,n\}$ equipped with the uniform probability measure. For a permutations $\sigma \in \Omega$, let $X(\sigma)$ denote the number of ...
2
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1answer
48 views

domino's pizza claim

I just got a dominos promotional flier through the post and one of the graphics advertising 'create your own pizza' lists the various toppings and claims there are 'more combinations than people in ...
2
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0answers
10 views

Noisy contraction mapping

I am trying to describe the behavior of the iterates of a contraction mapping when noise is added. Given a real valued random variable $X_{0}$ a sequence $\{Z_{n}\}_{n=0}^{\infty}$ of i.i.d real ...
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0answers
11 views

Localisation in the proof of Ito's formula

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows: Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a ...
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0answers
12 views

Markov Inequality for almost positive random variables

I came across the following problem in my research, where I want to apply Markov inequality to bound the tail probability of a random variable, X. However, the random variable X is not strictly ...
2
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0answers
13 views

Is this formula for disintegration correct?

I struggle to understand the concepts of 'regular conditional distributions' and disintegration. In this case, let $(X_t : t \geq 0)$ be a stochastic process (let's say it takes real values, for ...
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0answers
38 views

Is p(x)dx equal to dp(x)?

I'm confused with the definition of the expectation operator. Assume a random variable $X$ having a probability distribution $p(x)$. Then the expected value of $X$ can be computed as $\int xp(x)dx$. ...
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1answer
30 views

Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E[X^k] = E[Y^k]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the same distribution?
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0answers
18 views

find density of a distribution from the convolution [on hold]

if i have three distributions $\mu, g, p$ such that $\mu=g\star p$ (a convolution between g and p) then Is the density of $\mu$ given by $$h(x)=\int{g(x-y)p(dy)}$$? i am not clear about the ...
0
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1answer
29 views

Volume and Probability of a region given by a random variable

I am currently reading this paper. It is about nearest neighbors of a query point $X_q\in\mathbb{R}^k$ within a point set $P=\{X_i\mid X_i\in\mathbb{R}^k\}$, where the points have distribution $p(X)$ ...
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0answers
14 views

A question on the Kullback-Leibler divergence

Suppose $(Z,Y,X)$ are two random variables in $\mathbb{R}\times\mathbb{R}\times H$, where $H$ is some Hilbert space. I'm interested in the Kullback-Leibler divergence between the $P_{ZX}$ and ...
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0answers
10 views

Special case of bernoulli's trail

Lets assume that there are $n$ number of trains visiting $m$ stations. What is the probability that $k$ out of $n$ trains meet at $j_{th}$ station given that $P_{ij}$ is the probability of $i_{th}$ ...
2
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0answers
15 views

Poker dice probability of rolling 2 pairs

Poker dice are played by rolling 5 dice. Let A be the event of rolling 2 pairs. (e.g. 1,1,2,2,3.). Find $\mathbb{P}(A)$. So my answer is as follows: $$\mathbb{P}(A) = ...
2
votes
1answer
30 views

Exponential Tilting

Consider a random variable $Y$ with density function $f_Y(y)$ and moment generating function $m_Y(t)$ and cumulant generating function $\kappa_Y(t)$. Then a random variable $X$ derived from $Y$ by the ...
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0answers
12 views

Generating uniform permutations by a particular method

Let $A$ be a uniformly random permutation of the numbers $\{1,2,\cdots,n\}$. I want to generate a uniformly random permutation from $A$ on the numbers $\{1,2,\cdots,n,n+1,\cdots,n+m\}$. In other ...
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0answers
22 views

Probability Theory - Question on probability convergence and its probability (possibly law of large numbers problem)

Here is a problem that caught my curiosity. I am not sure of this, but is it some form of the Law of Large Numbers? Whether it is or not, I would be very interested to see the proof of it if anyone ...
2
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1answer
13 views

Expansion of cumulant transform

Verify the following expansion for a cumulant generating function of a random variable $X$. \begin{align} \kappa(t) & = \mu t + \frac{1}{2}\sigma^2t^2+\frac{1}{6}\rho_3\sigma^3t^3 + ...
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2answers
20 views

Equivalence involving expectation

I am stuck with the following problem, where I am asked to prove/disprove the following hypothesis: Is $\mathrm{E}\{e^{\max_i X_i}\} = \mathrm{E}\{\max\limits_i e^{X_i}\}$, where the $X_i$'s are ...
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2answers
23 views

How to take this exponentials

Given an expansion of a cumulant function as follows: $$ \kappa(t) = \frac{t^2}{2} + \frac{\rho_3 t^3}{6\sqrt{n}} + \frac{\rho_4t^4}{24n} +O\left(\frac{1}{n\sqrt{n}}\right), (*) $$ where ...
1
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1answer
30 views

Conditional independence of random variables

Let $(X_t),$ $(Y_t)$ be independent bounded martingales (for filtration $ \{ \mathcal{F}_t \}$ )which converge to $X_\infty$ and $Y_\infty$ respectively, by the martingale convergence theorem. Let $\{ ...
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1answer
14 views

Conditional expectations with different measures

Let $\mathbb Q \sim \mathbb P$ on ($\Omega, \mathcal F$) and $\mathcal G \subset \mathcal F$. We also have a Radon-Nykodym derivative $$ \rho = \frac{d \mathbb Q}{d\mathbb P} \bigg|_{\mathcal F} $$ ...
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1answer
6 views

formula to produce a set of probability distributions for a set of integers between a lower and upper bound with a given mean value

The goal is to establish a set of probabilities to be used to select an integer value where the probability of selecting I is Q, I+1 is R, I+2 is S, ... I+n is Z and such that the integer with the ...
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2answers
24 views

No pairs when drawing cards from deck

Suppose we are dealt five cards from an ordinary 52-card deck. What is the probability that we get no pairs (i.e. all cards are different values). I'm not sure if I've got the right answer on this ...
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0answers
28 views

Multiple absorbing boundaries

I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be ...
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2answers
39 views

Probability involving cards

Given a deck of 8 cards containing 4 jacks and 4 queens, if 4 cards are selected with replacement (I put the card back after I pull it) how can I calculate the probability that no consecutive cards ...
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0answers
17 views

Statics maximization

I really need a help to compute the following maximization problem. $$\max _{p(x), E(x) \leq \alpha} \int_x \int_y \frac{p(x,y)^2}{p(x)p(y)} dx \,dy$$ Suppose that : $$p(y|x)=\frac{1}{\sqrt{2 \pi ...
1
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1answer
24 views

Distribution of maximum of correlated Gaussians

Let $X_1,X_2,...,X_n$ be iid standard Gaussian random variables. Consider the set of random variables $M =\left\{\left( X_i-X_j\right) :i,j = \left\{1,2,\dots,n\right\} \& i\ne j\right\}$. I ...
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0answers
17 views

How to formulate a law of large numbers for a continuum of random variables?

Consider the random variables $I(t)=1$ with probability $p$ and $I(t)=0$ with probability $1-p$ for each $t \in T=[0,1]$. The $I(t)$ should be essentially pairwise independent. Sun and Zhang (2009, ...
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0answers
6 views

Is it interesting to define probability using a partial ordering?

I am studying probability for an actuary exam, but I remember the concept of a partial ordering from my combinatorics course. After thinking about it for a moment, I wondered if it would be possible ...
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0answers
10 views

p-variation of semimartingales

Does every (particularly continuous) semi-martingale have bounded 2+$\epsilon$-variation for all $\epsilon>0$? Note that I am not asking, whether they have bounded quadratic variation - that is ...
0
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1answer
16 views

Representation of a homogeneous Poisson process

Given a homogeneous Poisson process $\{N(t)\}_{t\ge 0}$ with intensity rate $\lambda>0$. Does someone know how to prove that there is a sequence of i.i.d. $Exp(\lambda)$-distributed random ...
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2answers
29 views

Pdf of scaled Nakagami?

I am trying to find the probability distribution function (pdf) of the following $$Y=a X $$ Given that $$X \sim \operatorname{Nakagami}(m,1)$$ $$a \,\text{positive constant}$$ Is the pdf of $Y$ ...
2
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1answer
27 views

Fubini's theorem for conditional expectations

I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ if finite then: $$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$ I just ...
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0answers
11 views

Measurability of a function $g(x):=\sup\{t:\ x(t)=0\}$ where $x$ are functions

THE PROBLEM: Let $g:\mathbb{R}_+^{\mathbb{R}_+}\to\mathbb{R}_+$ be defined as $g(x):=\sup\{t:\ x(t)=0\}$. Is $g$ measurable? MOTIVATION: If $X$ and $Y$ are random variables taking values in ...
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0answers
27 views

Conditional Probability and Events

Is A|B an event? If not, can we call P(A|B) the probability of A|B? Because Probability refers to a function whose domain is an event space. So it wouldn't make sense to talk about the probability of ...
2
votes
1answer
19 views

Calculating conditional expectation

I have to calculate $E\left(\int_1^4 W_t^3dt |\mathcal F_2\right)$ My solution: $E(\int_1^4 W_t^3dt |F_2)=E(\int_1^2 W_t^3dt |F_2)+E(\int_2^4 W_t^3dt |F_2)=\int_1^2 W_t^3dt+\int_2^4 E(W_t^3 |F_2)dt$ ...
2
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2answers
24 views

Exponential Families defined by Radon-Nikodym Theorem

Let $X \in \mathbb R^d$ be a random vector on space $(\Omega, \mathcal F, \mathbb P)$ and its Laplace transform $\varphi(\theta) := \int e^{\theta\cdot X(\omega)}\mathbb P(d\omega)$ exists for a row ...
0
votes
1answer
18 views

Covariance of random variable as a function of distribution of noise

Consider the following stochastic difference equation \begin{equation} x(t+1) = x(t) + \nu(t+1) \end{equation} where, $x(t)\in\mathrm{R}$ be one dimensional and $\nu(t)$ be the disturbance with an ...
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1answer
33 views

Existence of localizing stopping times that reduce a local martingale to a square integrable martingale

Something is weird from a proof that I am reading: The well-known theorem of characterization of quadratic variation states that: Suppose $X$ is a continuous local martingale and $A$ is a continuous ...
0
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1answer
29 views

t-student distribution

I've got this problem: Here, if $Z,W$ are independent random variables, and $Z$ has normal standart distribution and $W$ has $\chi^2$ with $n$ degrees of freedom, $T=\frac{Z}{\sqrt{\frac{W}{n}}}$. I ...
0
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1answer
24 views

Expectation of B(1) times stochastic integral?

I need to find the value of this expectation: $$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)$$ $B=(B(t))_{0\leq t\leq1}$ is a standard Brownian motion on $[0,1]$ and $f=(f(t))_{0\leq t\leq1}$ is ...
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0answers
32 views

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find the joint density.

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$. My attempt: Since the random vector is uniform it will have ...
1
vote
1answer
19 views

How to prove an inequality $\left| {g(j + 1)} \right| \le 5/4$ in Stein's method for Poisson approximation

The following is a lemma in Barbour, A. D., Holst, L., & Janson, S. (1992). Poisson approximation. Oxford: Clarendon Press,p7. For $j=1,2,...$ and $\lambda > 0$, we have $\left| {g(j + ...
1
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1answer
26 views

How to integrate over stochastic paths in stochastic calculus?

Suppose $X$ is a stochastic process with a certain probability distribution that is not time-dependent. $X$'s value is assumed to be a real number. Now we want to take the average of $X$ over every ...
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2answers
46 views

If $E[X\mid Y]=0$ a.s., does it follow that $E[X\mid f(Y)]=0$ a.s.? [on hold]

If $E[X\mid Y]=0$ a.s., does it follow that $E[X\mid f(Y)]=0$ a.s., for any measurable function $f$?
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2answers
22 views

Range of a random variable - measure theory

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Sometimes, the convention is used that a random variable is a map from a probabilty space to $\mathbb{R}$, but let's not adopt this. So let $X$ ...
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23 views

Is there some limit result with conditional expectation?

Let $Y$ be a random variable and $\mathcal F_n$ be an increasing sequence of sigma algebras. Do we have some limiting result for $E[Y \mid \mathcal F_n]$? For example, if $$\mathcal F_\infty = ...
2
votes
1answer
28 views

Resolvent of a Markov process

I have a question about theory of Markov processes. Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $E$ be a Hausdorff topological space and $\mathcal{B}(E)$ be its Borel $\sigma$-algebra ...
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1answer
23 views

Characterization of pre-orders

Let $X$ be an arbitrary set, and $\leq$ be a pre-order on $X$. Does there always exist $u:X\to \Bbb R$ such that $x'\leq x''$ iff $u(x') \leq u(x'')$? If this is not true in general, is that true ...
2
votes
1answer
23 views

Proof of mean and vector

Let $X_1,\ldots,X_n$ be a random sample from $N(\mu, \sigma^2)$. Show that the sample mean and each $X_i-\bar X, i= 1,\ldots,n$, are iid. Actually $\bar X$ and the vector $(X_1-\bar X,\ldots,X_n-\bar ...