# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

5,176 questions

**-3**

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23 views

### Mathematics grade 12 probability [on hold]

Stella has 12 coins in her hands, six of these coins are Looney. She accidentally (at random) drops three of these points. Determine the probability that exactly two of the drop coins are loonies.

**1**

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18 views

### The minimum of the reciprocals of some Poisson random variables

Let $X_1,\dots,X_k$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $k$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}}...

**-2**

votes

**0**answers

30 views

### Calculating the probability of obtaining exactly four distinct values when a die is rolled six times [on hold]

What is the probability of getting
a) 4 distinct numbers (no order in the outcome e.g. 1,2,3,4 or 4,5,6,2 etc)
b) 5 distinct from rolling a die 6 times
So far I was able to calculate the ...

**1**

vote

**0**answers

20 views

### Asymptotics for a random set cover problem

Suppose you are given a positive integer $k$ and a probability distribution $f$ on the positive reals. I am interested in the limiting behavior of the following process as $n\to\infty$:
Create an ...

**0**

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**0**answers

39 views

### Ladder times of a Brownian motion with drift

Let $(B_s)_{s_\geq 0}$ be a standard Brownian motion and fix $t>0$. For $u>0$, set $T_u=\inf\{s>0, B_s+s t>u\}$. Now consider $x>0$ such that $\sup_{0 \leq s \leq x} (B_s+st)=B_x+xt$ ...

**-2**

votes

**0**answers

53 views

### Birthday Calendar Gaps [on hold]

I work at a company that posts a birthday calendar. I noticed that there was a string of four consecutive days with no birthdays. What is the probability of that happening?
Problem Statement
Given n ...

**0**

votes

**1**answer

43 views

### Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...

**2**

votes

**0**answers

63 views

### Markovian Control in a stochastic control problem

I have a very simple control problem.
\begin{align*}
V(0,X) &= \sup_{(C_t)_t} E_0 \left[ |X_t| 1_{C_t = 0} + C_t \right] \\
\text{ s.t. } & d X_t = C_t d B_t, \quad X_t \in [-1,1], C_t \ge 0,...

**1**

vote

**1**answer

94 views

### Existence of certain event

Suppose that $X$ is an unbounded random variable such that $\operatorname EX=0$ and $\operatorname E|X|^q=1$ with some $q>2$. Only the distribution of $X$ matters, so the probability space can be ...

**0**

votes

**1**answer

42 views

### How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...

**7**

votes

**2**answers

803 views

### Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...

**0**

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**0**answers

71 views

+50

### Range of convergence for ratio of successive binomial tail probabilities

For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...

**5**

votes

**1**answer

136 views

### Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$

If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...

**-5**

votes

**0**answers

34 views

### Is the effect of Tr on Y identifiable conditioning on G? [closed]

enter image description here
I wonder if Tr and Y are independent conditioning on G? I am working on a causal inference problem, and I wonder whether and why the effect of Tr is identifiable if G is ...

**0**

votes

**0**answers

25 views

### p-Variation distance defines semi-martingales

Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...

**8**

votes

**2**answers

502 views

### Random Walks on high dimensional spaces

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), ...

**1**

vote

**1**answer

34 views

### Jump size of process in proof of martingale CLT bounded

I'm currently trying to understand the proof of the final theorem of (Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time) where I need the following:
For $\{...

**5**

votes

**1**answer

165 views

### Relation between the two possible KL divergences of two distributions

Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $D\left(Q\parallel P\right)$ in terms of an upper bound on it?
Also, given this upper bound on $D\left(P\parallel ...

**1**

vote

**1**answer

112 views

+50

### Time discretization in the Feynman-Kac formula with boundary conditions

I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...

**2**

votes

**0**answers

109 views

### Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...

**3**

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**1**answer

67 views

### Powers of Frobenius norm of sum of random matrices

For $i= 1, \ldots, n$, let $A_i \in \mathbb{R}^{d \times d}$ be random i.i.d. matrices with $E [A_i] =0$.
Can we relate (upper bound) $E[\|\sum_{i=1}^n A_i \|_F^4]$ to $E[\|A_i\|^4_F]$ ?

**6**

votes

**1**answer

170 views

### Approximating $\mathbb{E}[1/X]$

I am well aware (as for instance discussed here https://math.stackexchange.com/questions/910846/is-it-true-in-general-that-e1-x-1-ex) that for an arbitrary random variable $X$ it does not hold that $\...

**4**

votes

**1**answer

105 views

### Is there a counterexample to the Thin Shell Conjecture for sub-exponential distributions?

The thin shell conjecture states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies
$$\mathbb{P}\...

**3**

votes

**0**answers

72 views

### Upper bounding the start of a distribution's CDF, given bounds on first moments

Take nonnegative random variables $X$ whose first $K$ moments have bounds:
$\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$.
In this case what is an upper bound for $P(X\leq O(\mu))$?
I am ...

**4**

votes

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79 views

### On a much weaker version of the Normal conjecture

I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...

**3**

votes

**1**answer

109 views

### Expected size of the smallest preimage set

Let $f$ a function from $\{0, 1 \}^{2n}$ to $\{0, 1 \}^{n}$ uniformly picked at random. I would like to have an estimation of the expected size of the smallest premiage set of $f$, more formally $\...

**0**

votes

**1**answer

43 views

### Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$

Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$.
Question
Given $\epsilon > 0$ (may be assumed to be very small), what is ...

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votes

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25 views

### Tail Probabilities of certain subsets of rolls of a fair die [closed]

We have a fair n-sided die and we are interested in rolls of a particular subset of the die, say [1,m] with m < n. We roll the die a number of times and observe that b of the sides in [1,m] have ...

**2**

votes

**0**answers

78 views

### The Kleisli Category of the Monad of Measures of Finite Support and its composition formula

In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. Let's call this ...

**-1**

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32 views

### A limit for two correlated variables [on hold]

Suppose we have two correlated Normal variables $X_A$ and $X_B$, with respective standard deviation $A$ and $B$, and correlation $\rho$. The variable $X_A + X_B$ has a standard deviation in excess (...

**2**

votes

**1**answer

99 views

### bp continuity of Markov operators / semigroups

Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...

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66 views

### mean distance between subspaces

Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...

**2**

votes

**1**answer

99 views

### How to uniformly sample a square (0,1)-matrix whose trace is 0 and whose row sums and column sums are the same?

Happy New Year!
Suppose I would like to sample a $n \times n$ (0,1)-matrix whose trace is 0, and whose row sums and column sums are all $m$ with $1 \le m \le n-1.$ How can I sample this matrix ...

**4**

votes

**1**answer

66 views

### Sampling uniformly from the vertices of a polytope

I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...

**0**

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55 views

### Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile

$F(x)$ and $G(y)$ are distribution functions.
Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as
$$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$
and
$$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...

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61 views

### Is there solution to a backward stochastic differential equation with $yz$ in the generator?

Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...

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**0**answers

71 views

### What are the open problems in rough volatility models?

The 2014 paper Volatility is Rough argues that empirically, fractional Brownian motion with $H=0.1$ is a good description of volatility that comes out of high frequency trading.
Since then there has ...

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**1**answer

116 views

### What's the probability of two independent events in time domain?

Suppose there are two independent events A and B. The probability that A or ...

**2**

votes

**1**answer

132 views

### Fourier transform of a simple random walk

Consider the usual simple random walk on $\mathbb{Z}$, taking steps of +1 or -1 with equal probability. Of course, each trajectory corresponds uniquely to an element of $\{-1,1\}^\infty$. Now, there ...

**0**

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**0**answers

41 views

### Limiting a sequence of moment generating functions [migrated]

I was trying to solve the following problem:
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...

**0**

votes

**0**answers

56 views

### Binomial CDF - Wiki vs Reality [migrated]

In https://en.wikipedia.org/wiki/Binomial_distribution the binomial CDF is defined as
$F_{Bin(n,p)}(k) = I_{1-p}(n-k, k+1)$,
Where $I$ is the regularized incomplete beta function.
There is a proof ...

**3**

votes

**0**answers

118 views

### Are there any conditions on the moments that make a measure a probability measure?

For a positive Borel measure $\mu$ on the real line interval $[-1, 1]$, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions ...

**5**

votes

**1**answer

80 views

### What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?

Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.
Let
$$p(x) \propto \exp(-\|x\|_1/\sigma)$$
be the pdf of the Laplace ...

**8**

votes

**1**answer

153 views

### On the existence of a particular type of finite measure on $\mathbb N$

Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...

**6**

votes

**0**answers

64 views

### Distributions of “sequential” binomials

I have come across the following stochastic process which seems very elementary, although I do not know any appropriate terminology for it; I greatly appreciate any suggestions!
Suppose I am given ...

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**1**answer

64 views

### Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$

Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$.
Question
What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...

**2**

votes

**1**answer

108 views

### About a pattern of hitting times for a simple random walk

Let $\omega_1, \omega_2, \ldots$ be uniform iid on $\{-1,1\}$, and let $X_n = \sum_{i=0}^n \omega_i$ be the corresponding simple random walk. Fix some integer $N$, and let $h^+_N$ be the first time $...

**5**

votes

**1**answer

58 views

### Estimating the size of the remainder in a random partition

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...

**1**

vote

**0**answers

51 views

### Almost orthogonality of independent random vectors [closed]

If $X_1$ and $X_2$ are two independent isotropic random vectors in $\mathbb{R}^n$,
then $\mathbb{E}\|X_i\|_{2}^{2}=n$, $\mathbb{E}\langle X_1,X_2\rangle^{2}=n$.
How can I show from the above result ...

**0**

votes

**1**answer

32 views

### Minimizer for Mean-Variance Portfolio Optimization [closed]

Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set
$$
\{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ?
$$