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Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.

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Let $\{P_\vartheta \mid \vartheta \in \Theta\}$ be a family of probability measures on $(\mathbb R^n,\mathfrak B_{\mathbb R^n})$ dominated by some $\sigma$-finite measure $\mu$. We say that $\{P_\...
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I'm trying to solve a problem from the book Shiryaev A.N. Problems in Probability. I solved it, but at the end I noticed that in the definition of $\Sigma_k$ we have the expression $\sum_{i=0}^k x_i = ...
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Let $\mathcal{A}$ and $\mathcal{B}$ be family of subsets of $X$, and let $x_1, \ldots, x_n \in X$. Denote $x=(x_1, \ldots, x_n)$ and $I_A(x) = (I_A(x_1),\ldots, I_A(x_n)) \in \{0,1\}^n$, where $I_A$ ...
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I am just getting started on Tao's Topics in Matrix Theory, so I am still on the review of probablity theory. Exercise 1.1.7.ii says: If X has finite second moment, show that $$ M(X) = E(X) + O(Var(X)^...
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I am reading the original paper for the KPSS test "Testing the null hypothesis of stationarity against the alternative of a unit root", and equation 14 says $\hat{\eta_\mu} \rightarrow \int_{...
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Problem setup Suppose we have a sequence of $N$ numbers, and generate pseudo-random sequences by sampling from these $N$ numbers. The number of possible sequences with $k<N$ distinct elements is ...
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Let $(x,y)\in\mathbb{R}^d\times\{0,1\}$ with rare positives $\pi=\mathbb{P}(y=1)\ll1$. For a linear score $s_w(x)=w^\top x$ and $K=K_n$ with $K_n/n\to0$, define $$ \mathrm{Prec@K}(w)=\frac1K\sum_{i\in ...
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Pricing an European Call Option (Binomial Lattice Model): Why insist on using the Expected Value when it is not the representative path over time? Intro______________ I am self-studying financial math ...
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The following is from the book High-Dimensional Statistics: A Non-Asymptotic Viewpoint by Wainwright. We define for a point $x^n_1 = (x_1, \ldots, x_n)$, $\mathcal{F}(x^n_1) = \{(f(x_1),\ldots, f(x_n)...
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When working with a set of elements, one may wish to identify a subset whose variance does not exceed a given threshold. One possible approach is to examine smaller subsets first; if these subsets ...
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My knowledge level is as much as a high-schooler. I know some limits, algebra, statistics and probabilities. Can one prove this is true if probability of each event is equal, and $A$ and $S$ are ...
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Whenever we calculate the mean of a given data set, we first calculate the class marks, then multiply it by the frequency then divide it by the total frequency. But, what if, instead of assuming the ...
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I was looking through some old notes I wrote on Markov Chains, and I included a proof of the limiting behavior of an irreducible, aperiodic Markov chain. Looking back at it, I saw a gaping hole in the ...
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Consider samples $\{x_1, \ldots, x_n\}$ of some random distribution. Let $\mu = \frac 1n \sum_i x_i$ define the mean, $\sigma^2 = \frac 1n \sum_i (x_i - \mu)^2$ be the variance and $\gamma^3 = \frac 1 ...
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Suppose that $\mathbf{x}$ is normally distributed, $\mathbf{x} \sim \mathcal{N}(\boldsymbol \mu, \boldsymbol \Sigma)$. Under the transformation $\mathbf{h} = \boldsymbol\Sigma^{-1} \mathbf{x}$, we ...
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I am interested in sampling fractal like functions. In three dimensions, but for now let's focus on the real line case. Smooth case If I have a smooth function $f$ I can generate a sampling of $f$ by ...
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Suppose $X_1, \cdots, X_N$ are realizations following $N(\mu, \sigma^2)$ and $i\in [N]$. Suppose further that we group realizations $X_i$ into index set $S$ if $s < X_i < t$ for scalars $s, t$. ...
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Rules of the Game: You play with a fair 6-sided die. You choose a target number, for example: "2". Then you repeatedly roll the die, one throw at a time. If you roll a 2 at any point, the ...
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The following is from: High-Dimensional Statistics: A Non-Asymptotic Viewpoint Let $\mathcal{F}$ be a class of functions $f \colon \mathcal{X} \to \mathbb{R}$ and let $(X_1, \ldots, X_n)$ be drawn ...
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For a positive random variable $X$, the entropy is defined as $H(X) = \mathbb{E}(X \log X) - \mathbb{E}(X) \log (\mathbb{E}(X) )$. I want to prove following variational representation: \begin{align*} ...
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There are $n=1000$ voters who have to vote for one of two candidates. Suppose $n_1$ voters support candidate 1 and $n_2 = 1000-n_1$ voters support candidate 2. I do not know $n_1$ and $n_2$; in order ...
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Consider an Ito Stochastic Differential equation of the form $$dX=f(X,t)dt+g(X,t)dW ,$$ with $dW$ a real Wiener process. Under which conditions will $X$ and its higher moments $X^m$ satisfy the ...
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Let $(\mathcal{Z}, \Sigma)$ be a measurable space, and let $\mathcal{P}$ be a family of probability measures on it. For each $P \in \mathcal{P}$, let $Z = (X, Y)$ be a random variable with law $P$, ...
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Could these formula been accurate estimators of the Median Path of a Binomial Lattice Model? In the last added section I believe I almost show that the Expected Geometric Growth is actually the Median ...
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For non-negative random variable $Z \geq 0$, let $H(Z) = \mathbb{E}(\log Z)- \mathbb{E}(Z) \log \mathbb{E}(Z)$. In High-dimensional statistics by Wainwright, there is the following lemma. Lemma (...
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Suppose $X$ is a random variable with $\mathbb{E}\left[X\right] = \mu $ and $\mathrm{Var}\left( X\right) = \sigma^{2} $. For which value of $a>0$ is the value of $$\mathbb{E}\left[\left(aX - \dfrac{...
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To be more precise: Imagine I had a probability distribution $p(x;a)$, where $x\in \mathbb{R}$ a result and $a$ a parameter. If $a$ was fixed, I can empirically determine $p(x;a)$, by drawing infinite ...
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I have a k-dimensional vector A of random variables, where each random variable $A_i$ follows a normal (Gaussian) distribution with mean $m_i$ and standard deviation $c_i$. So, the distribution of A ...
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I am puzzled by a question on the statistics complex hypothesis testing, namely, there is a difference between particular theoretical PDF of a statistic and the simulated one, and I have no idea why ...
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I'm trying to get a deeper understanding of Principal Component Analysis (PCA), and I keep coming across the point that we must center the data around zero before determining the principal components. ...
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Consider the order $3$, rank $2$ tensor $$ T = 2e_1^{\otimes 3} + 5(e_1 \otimes e_2^{\otimes 2}) \in \operatorname{Sym}^3(\mathbb{R}^2) $$ This is rank $2$ in the sense that it is written as a sum of $...
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Let $X_1,X_2,\dots,X_n$ be iid random vectors in $\mathbb R^d$ with covariance matrix $\Sigma$. I want to show that $$\hat\Sigma := \frac{1}{n-1}\sum_{k=1}^n(X_k - \overline X_n)(X_k - \overline X_n)^\...
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We will say that a zero-mean random variable $X$ is $(\nu, \alpha)$ sub-exponential if for all $\left|\lambda\right|<\frac{1}{\alpha}$, $\mathbb{E}(e^{\lambda X}) \leq e^{\frac{\lambda^2 \nu^2}{2}}$...
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What is the benefit of using the force of mortality $μ_x$ instead of the Probability Distribution Function $f_x(t)$ of a lifetime random variable $T_x$? I understand that if I have the continuous ...
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In logistic regression, is there a FUNDAMENTAL reason why I map a continuous value to a probability? Couldn't I simply define a decision threshold from a continuous value? What is the mathematical ...
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Why is it incorrect to approach the problem like so, relying on multiplication rule: For the 3 of a kind Choose any card at random: 52 choices (52). Once that's locked in, there are 3 choices (3) to ...
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first of all I wanted to say I haven't red statistics in a proper order, I'm preparing for an olympiad, so pardon me if it's too obvious. The main defination of memoryless property as I know is: $P(X&...
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Let's say there are $n$ voters who are going to vote to one of two parties A and B. We want to estimate in advance what party is going to win, so we poll some $k\ll n$ citizens at random. If party A ...
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Ann and Jim are about to play a game by taking turns at tossing a fair coin, starting with Jim. The first person to get heads twice in a row will win. (For example, Ann will win if the sequence (...
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Let $\{X_i\}_{i=1}^n$ be a sample of i.i.d. random variables with compact support $[0,1]$, $H$ be a symmetric kernel around 0, that integrates one, with compact support $[-1,1]$, and $h_n$ a function ...
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Imagine that you observe events occurring at random times. More precisely, the time intervals between events are IID and drawn from the distribution described by the p.d.f. \begin{align} \rho(t;\tau,...
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There is "standard Galton board" where particles form normal distribution. Probability distribution for normal distribution is widely known. Does anybody know how to calculate a probability ...
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Given A Bernoulli trial $f$ which yields a success with probability $p$ (in my specific case, $p \approx 2^{-20}$) A required number of successes $S$ (typically around 10^5 in my case) A number of ...
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It may be a beginner-level question, but it has been genuinely bugging me. Suppose we have two vectors on the unit sphere, defined as follows: $$ v_{1} = \left( \sin \phi_{1} \sin \gamma_{1}, \sin \...
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I’m working on the following exercise: Exercise: Let $g$ be a positive integrable function defined on $(0, \infty)$. Define the probability density function \begin{align} f_{\theta, \eta}(x) = \begin{...
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I am curious about the probability distribution for when half of the 'atoms' were to 'decay', which happens when at least half of the striated disks lose a billiard. I am not saying anything about how ...
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Suppose I measure 100 samples of a normal distribution and use them to compute a standard deviation. Is there a way to compute +/- error bounds on my computed mean value for standard deviation if I ...
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There is a dice game called "Can't Stop Express" with a basic principle: you roll 5 dice after each roll, you organize the dice in 2 pairs and a "5th die" During the game ...
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Setup. Let $p\in[0,1]$ be a continuous random variable with density $f(\cdot)$. Assume that $f$ is bounded, continuously differentiable, and has full support $[0,1]$. Let $a_1$ and $a_2$ be distinct, ...
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Let $u\sim P_u$, $v\sim P_v$ be two random elements taking values in a separable Hilbert space $H$. If there is a Borel set $D$ such that $P(u\in D) =P(v\in D)=1$, is it true that $P_u=P_v$ if and ...
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