Questions tagged [probability-theory]
For questions solely 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 [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.
43,531
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Asymptotic distribution of $T_{n}=2n(\sqrt{\bar{X}}-\sqrt{\bar{Y}})^{2}$
Let be independent Poisson random variables with $X_{1}, ..., X_{n}\sim Poisson(u)$ and $Y_{1}, ..., Y_{n}\sim Poisson(v)$. To contrast the hypothesis testing $H_{0}:u=v$, propose the next statistic:
$...
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0
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35
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Convergence of the sum of exponential random variables
Let $\{X_n\}$ be a sequence of nonnegative random variables such that for each n, $X_n$ has density $\lambda_n e^{-\lambda_n x}$ for $x \geq 0$ and $\lambda_n > 0$
i) Show that if $\sum_{n=1}^\...
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Can I think of probabilities as proportions instead?
I am new to probability theory, so bare with me if I do not nail all of the terminology (I will still try my best)! Also, I gave a short "What is my question" sentence, but I invite you to ...
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Let $(\Omega,F)$ be a measurable space, $F=\{\varnothing,\Omega\}$, prove that $X:\Omega\to\Bbb R$ is a random variable iff $X$ is constant.
Let $(\Omega,F)$ be a measurable space, $F=\{\varnothing,\Omega\}$ prove that $X:\Omega\to\Bbb R$ is a random variable if and only if $X$ is constant.
I was thinking this is not necessarily true as if
...
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How can I show that a Poisson process with my definition below has stationary and independent increments?
We had the following definition:
Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, \Bbb{P})$ be a filtered probability space. An $(\mathcal{F}_t)_t$ Poisson process $(N_t)_{t\geq 0}$ is a right ...
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Calculating a probability with the hypergeometric distribution
I'm studying the hypergeometric distribution with the probability mass function
$$
Pr(X=k) = \frac{{K \choose k}{(N-K) \choose (n-k)}}{{N \choose n}},
$$
where $N$ is the population size, $K$ is the ...
2
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1
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40
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Trouble understanding order statistics
Order statistics were introduced in my text as follows:
I am trying to understand what this means.
$X_1 , \dots , X_n$ is a random sample, i.e. an independent and identically distributed sequence of ...
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help with probability theory question
Let $v$ be a probability measure on $\mathbb{R}$ such that $v$ has no atom. Let $\left(x^{i, N}\right)_{1 \leq i \leq N}$ be the sequence of real numbers defined by:
$$
\begin{aligned}
x^{1, N} & \...
2
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1
answer
29
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Nonincreasing cdf? Potential error in my textbook..
Image from Introduction to Mathematical Statistics (7th edition) by Hogg, McKean & Craig:
I hope I'm not missing something obvious here, but isn't a cdf supposed to be nondecreasing? If $F$ is a ...
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why eigenvalues distinct follows for gaussianorthogonal ensemble
the joint law of entries of a Guassian orthogonal ensemble matrix $G O E_n$ is given by
$$
\mathbb{P}\left(\sqrt{n} \mathbf{X}_n \in B\right)=2^{-n / 2} \pi^{-n(n+1) / 4} \int_B e^{-\frac{1}{2} \...
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1
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How does $\Omega$ figure in stochastic processes?
So I read this page for clarification on trajectories and $X(\omega, \cdot): T\to \mathbb R$ maps while going through lectures on stochastic processes. I still have doubts which are described as ...
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Is Memorylessness a necessary assumption in Shannon's Proof of the Channel Coding Theorem?
It is often said that the achievability proof for Shannon's coding theorem relies on the channel being discrete and memoryless. At the same time, following the classical proof (using random coding and ...
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25
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necessary and sufficient condition about weak convergence on N (the set of natural numbers)
I would like to show that if all natural number $k$ satisfy that
$\displaystyle\lim_{n\to\infty}\mu_{n}(\{k\})=\mu(\{k\})$, then $\{\mu_n\}_{n=1}^\infty$ converges on $\mu$ weakly, where $\{\mu_n\}_{...
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Watanabe characterization of a Poisson process
There is an implication that I am not able to find by myself in my lecture notes.
I consider $X_t$ a cadlag process with values in $\mathbb{R}_{+}$ such that $X_t$ is locally integrable (with respect ...
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i need a stochatic differential equation that its solution will represent this data of wind direction in radians.
enter image description here
hi everyone
I have this data of wind direction in radians from -pi to pi (picture is added).
any idea which stochastic differential equation (SDE) can represent it? i need ...
1
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39
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Proof of weak convergence when domain of rv is N
I need to show:
If $X_i, i\geq1, X$ are random variables with domain $\mathbb{N}$, then $ X_n \rightarrow X$ weakly iff $\forall i \in \mathbb{N}: P(X_n = i) \rightarrow P(X=i)$.
The direction $\...
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49
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Sub-exponential Norm of Normal Distribution
Let $X \sim \mathcal{N}(0, \sigma^2)$. Define the sub-exponential norm as follows for a random variable $Z$:
$$ \|Z\|_{\psi_1} := \inf \left\{k>0 \vert \mathbb{E}\left[\exp \frac{|Z|}{k}\right] \...
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38
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$f(x,y)\stackrel{\text{a.s.}}{=}\frac{\partial^{2}F}{\partial x\partial y}(x,y)?$
The random variables $X$ and $Y$ with the joint distriution $F(x,y)$ possess a density $f(x,y)$(with respect to the Lebesgue measure) if for any $x,y$ $$F(x,y)=\int_{-\infty}^{x}\int_{-\infty}^{y}f(u,...
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Prove that X is not always random variable in case that X^2 is random variable [closed]
Could someone give me an idea how to prove it?
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. Suppose 𝐴 and 𝐵 are two events. Prove that 𝑃(𝐴) + 𝑃(𝐵) − 1 ≤ 𝑃(𝐴 ∪ 𝐵) ≤ 𝑃(𝐴) + 𝑃(B)
I'm totally confused on this since the only available proof is 𝑃(𝐴) + 𝑃(𝐵) − 1 ≤ 𝑃(𝐴 ∪ 𝐵) only explained in online.
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Find distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$
Let $X_{1},...,X_{n}\sim f(x)=Kx^{2}e^{\frac{−x^{2}}{2σ^{2}}}$ i.i.d. I need find the distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$. To do this, calculate the normalizing constant, K, ...
0
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1
answer
34
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Maximum of Sub-Exponential and its Tail Probability
Consider zero-mean sub-exponential random variables $\{X_1,...,X_n\}$ (not necessarily independent) with parameters $(\nu, \alpha)$. That is,
$$\mathbb E[\exp\{\lambda X\}] \le \exp\{\nu^2\lambda^2/2\...
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0
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18
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Multivariate method of moments
In Mathematical Statistics: Basic Ideas and Selected Topics (Kjell A. Doksum, Peter J. Bickel), problem 2.1.17 defined "multivariate method of moments" like this:
Then in problem 2.2.29(b), ...
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0
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25
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Conditional expectation (with average errors)
In Mathematical Statistics: Basic Ideas and Selected Topics (Kjell A. Doksum, Peter J. Bickel), problem 2.2.29 (a) asks:
Let $\epsilon_1,...,\epsilon_{n+1}$ be i.i.d. with mean zero and $Y_i = \mu + \...
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1
answer
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Conceptual doubt about the use of normal distributions
Let's say we have a set of scores from an exam that can range from 0-10. Given N=30 scores we can compute the mean ($\bar{x}=4.8$) and the standard deviation ($\sigma_{x}=2.0$). If we use the normal ...
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1
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120
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Proof of Kolmogorov's $0$-$1$ Law in Shiryaev
I know this theorem has been discussed in a lot of posts on this site however I was unable to find an answer that deals with the particular proof I am trying to understand. I'm interested in the proof ...
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47
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Variation of St. Petersburg Paradox
I was discussing the the St. Petersburg paradox and the following question came up:
Suppose the game doesn't end within nine rounds, then the player directly receives $2^{10}$ dollars , while ...
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1
answer
30
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Distribution of an estimator
Let $X_1,\ldots,X_n$ be indenpendent identically distributed random variables with density $$f_X(x)=\theta(1+x)^{-1-\theta}$$ for some $\theta>0$ and $x>0$. I would like to know how to compute ...
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Finding marginals of the uniform distribution on a triangle
The statement is:
Random vector $(X, Y)$ is uniformly distributed on a triangle $A = (0, 0)$, $B = (2, 0)$, $C = (1, 1)$. Find distribution and density functions of $X, Y$. Check if $X, Y$ are ...
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49
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help in understanding steps in product measure question
Let $v$ be a probability measure on $\mathbb{R}$ such that $v$ has no atom. Let $\left(x^{i, N}\right)_{1 \leq i \leq N}$ be the sequence of real numbers defined by:
$$
\begin{aligned}
x^{1, N} & \...
1
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0
answers
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+100
Independence is preserved by joint weak convergence
Suppose a sequence of random vectors $(X_n,Y_n)$ converges jointly to some $(X,Y)$ in the weak topology. Question: If $X_n$ and $Y_n$ are independent for all $n$, are also $X$ and $Y$ independent?
...
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Find this expected value [closed]
Let $\xi_1,\xi_2,\cdots, \xi_n, \cdots$ be a sequence of identically distributed continuously randomly variables and $\nu = \text{min}\{k: \xi_{k-1} > \xi_k\}$. Find $\mathbb{E}(\nu)$
I'm ...
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1
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Find the least sigma algebra that induces a real random variable
Consider $\Omega=\{1, 2, 3, 4 \}$. I want to find the least $\sigma$-algebra over $\Omega$, call her $\mathcal{F}$, so that $X(w)=w+1$ is a real random variable.
We want that, for any $A \in \mathcal{...
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1
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Verify that the application is a random variable (real)
Let $\mathcal{F} =\{\emptyset, \{1 \}, \{2, 3 \}, \Omega \}$ with $\Omega=\{1, 2, 3\}$, for $A \in \mathcal{F}$ we define:
$$\left\{\begin{matrix}
1 & \text{if $w\in A$}\\
0 & \text{if $w\...
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0
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Why the function $x\mapsto \nu(E_x)$ is measurable for $E_x$ is the section set of $E$ in a product space
Let $(X,\mathcal X, \eta)$ and $(Y,\mathcal Y, \nu)$ be measure spaces and denote $(X\times Y, \mathcal X \otimes \mathcal Y, \eta \times \nu)$ be the usual product measure space. I assume that they ...
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Marginal Conditional Expectation
Let $X: \Omega \rightarrow [0,1]^K$ be a random vector. Further, we assume that there exist a random vector $Z$ such that $\mathbb{E}[Z|X = x] = 0$ for all $x$ in the support of $X$.
Given this ...
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From cumulants of a random variable to its moments
This question is very related to:
Infinite sum of Gamma random variables with same shape parameter but different rate parameter
In particular I know that a random variable $Q_n'$ has $j$-th cumulant ...
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Can Central Limit Theorem say anything about Probability Density Function?
For a sequence $\left( X_1, X_2, \dots \right)$ of independent and identically distributed random variables with zero expectation and unit deviation, CLT implies
$$
\lim_{n\to\infty} P\left( \frac{1}{\...
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Harmonic Measure is absolutly continuous with respect to Lebesgue measure.
currently I am reading Greg Lawler's book conformally invariant processes and I am already having some experience with harmonic measures and calculating explicit Poisson kernels and related exit ...
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61
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Project Euler Problem 783: Expected value of sum of squares
I am trying to solve Problem 783 on Project Euler. I used a dynamic programming approach calculating the expected value given
a specific number of white and black balls. Then E(n_w, n_b) (where n_w ...
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35
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Approximation in terms of outer measure
This is Exercise 15 from Tao's note: https://terrytao.wordpress.com/2015/10/12/275a-notes-2-product-measures-and-independence/comment-page-2/#comment-682195. Given any subset ${E \subset [0,1]^A}$ (...
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Permuting function value at some set of points change Lebesgue integral?
Consider a space $(\Omega, \mathcal{F}, \mu)$ with sigma algebra $\mathcal{F}$, and a sigma finite measure $\mu$. I have a measurable function $f: \Omega\to \mathbb{R}$, where the codomain is equipped ...
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What is a wasserstein barycenter?
I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. ...
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2
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Applying LOTUS on the unconditional expectation of a conditional random variable
We have that
$$X \sim\mathcal U(0,1)$$ and given X = x then $$Y\mid X=x \sim \mathcal{Binomial}(n=5,p=x)$$.
Given that $E[Y] = 5/2$, I am asked to calculate
$E[Y^2]$.
I have decided to use the ...
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1
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0
answers
17
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waiting time between consecutive detections
I am studying a scenario where (N) particles are emitted at time (t=0), and the detection time since the origin follows an exponential distribution with time constant (L). Specifically, the PDF is ...
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0
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25
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Example of statistical data with the property of being contextual that is generated by quantum mechanics
Definition for the specifics of the question as well as an example of contextuality in quantum mechanics
I have a set of measurements acting on a 2 qubit state for whom the statistics of the ...
0
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0
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13
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Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
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66
views
Brownian motion and Holder-$\frac{1}{2}$-continuity
Let B be a Brownian motion. For every $K>0$, we have
$$
P[\inf \left \{ t>0: B_t\geq K t^{1/2} \right \} =0]=1 \quad\quad\quad(1)
$$
To prove this in Example 21.16 of Probability Theory (3rd ...
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1
answer
41
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Show that $\mathbb{E}(X) = \int^\infty_0 \mathbb{P}(X\geq x) dx$
I am learning probability theory and I have trouble going from $\int_\Omega X d \mathbb{P}$ to anything with the Lebesgue measure. How can I show that $\mathbb{E}(X) = \int^\infty_0 \mathbb{P}(X\geq x)...
0
votes
1
answer
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Proving this formula using Layer-cake representation
Let $K > 0$ be a fixed positive parameter and let us consider the set $\Sigma_{N} := \{-1,+1\}^{N}$, for $N \ge 1$. Given two elements $x = (x_{i})_{i=1}^{N}, y= (y_{i})_{i=1}^{N} \in \Sigma_{N}$, ...
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