Questions tagged [statistics]

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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Consistency of bootstrap estimator that is continuously $\rho_\infty$-Frechet differentiable

Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. ...
reyna's user avatar
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Complete Sufficient Statistis with pdf $f(x;\theta) =\theta(1+x)^{-(1+\theta)}$

[Bain Problem 31 p.356] Let $X_1,X_2,\cdots,X_n$ be a random sample of size $n$ from a distribution with pdf $$f(x;\theta) = \left\{\begin{array}{rr} \theta(1+x)^{-(1+\theta)}, &x>0 \\ 0, &...
Wildan B. W.'s user avatar
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Completeness of $X$ with law $N(\mu, \mu^2)$: Integral of Scale of a Function

Suppose that $f:\mathbb{R} \to \mathbb{R}$ is a Borel function such that, for every $a > 0$, we always have: $$\begin{equation*} \int_{\mathbb{R}}f(ax)\exp \left(-\frac{1}{2}(x-1)^2\right)dx =0 \...
温泽海's user avatar
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What's the proper error metric for a 'guessed' result when comparing with the observation data?

Hi Statisticians/Mathematicians, I am trying to compute the error of a 'guessed' value against some observation data, and am wondering what's the right process/approach. For example, there is a group ...
Edamame's user avatar
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Linear Combinations of Sample Mean Difference

What is the standard error of the mean difference between 2 variables (a & b)? I have the following data: $$ \sigma_{A} = 1.813529 \nonumber \\ \sigma_{B} = 1.932183 \nonumber \\ cov(A,B) = 2....
joelleoqiyi's user avatar
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Find the covariance [closed]

X1, X2, . . . , Xn ind from a cont. dist n which is symmetric around 0 (ie., −X1 d = X1 ). Suppose that E |X1| = 2. Find the covariance between y = X n i=1 Xi and z = X n i=1 I(Xi > 0). I(X1>0) ...
Sayantan Sinha's user avatar
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what happens to this joint mgf when$t_{1}$ and $t_{2}$ are arbitrarily small but not $0$

I have a question basically i am trying to calculate the E(XY) from the joint mgf. I found the joint mgf to be $$ -θ^2(-θ t_2 + θ^2 t_1t_2)^{-2} + (2 θ^2 t_2(-θ + θ^2 t_1)(-θ t_2 + θ^2 t_1 t_2)^{-3} $$...
princessfrostine's user avatar
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Likelihood ratio test of Poisson distribution

I'm given this problem: Let $X_1,...X_{100}$ be a random sample from a Poisson distribution with mean $\lambda$. Consider testing the hypothesis $H_0$: $\lambda=1$ vs $H_1$: $\lambda<1$. Consider ...
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Finding UMVUE for Correlation Coefficient in Multivariate Normal Distribution

It is described in Wikipedia that: In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance ...
Hank Wang's user avatar
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AIC for a Combined Model

Lets say I have a some variable C, which has two components with weights that sum to 1. For example A $\times$ 0.6 + B $\times$ 0.4 = C I want to compare two modelling approaches. Approach 1 involves ...
Michael Jones's user avatar
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How to find the variance of a biased estimator of variance of standard normal distributed random variable?

Let $Y_i$ be an independently and identically distributed standard normal random variable. Denote $S_1^2$ as estimator of $\text{Var}(Y_i)$. How to find the variance of the biased estimator $S_1^2$? $$...
Ermaolaoye's user avatar
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Probability of Different Outcomes Being First Given Base Probabilities in a Random Order

I'm given a set of probabilities p1, p2, ..., pn. corresponding to different outcomes. They are then randomly ordered and tested until one returns true or all are false. I want to find the probability ...
Brendan Clark's user avatar
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Is every method of moments estimator (MME) asymptotically normal distributed?

As already written in the question, I am asking myself whether every method of moments estimator (MME) is asymptotically normal distributed? Formulated differently, is every (central or non-central) ...
asdf1234's user avatar
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Calculate the posterior distribution

How can I solve the letter (a)? Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from $1$ to $N$. You see a cable car at random; it is numbered $203$. You ...
Siqueira's user avatar
2 votes
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Bayesian Inference Intractability

When looking at Bayesian posteriors $$ p(z \mid x) = \frac{p(x \mid z)p(z)}{\int p(x \mid z')p(z')dz'} $$ The denominator commonly intractable. I understand this is due to the possibility of high ...
Lehmann's user avatar
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Question about the null fraction parametrization for a poisson model (Mathematical statistics)

Im trying to understand the following in my mathematical statistics textbook: For a family of Poisson distributions it can be parametrized through its mean \begin{align*} E_{\lambda}(X)=\sum_{x=0}^{\...
xexon123's user avatar
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What is "implied standard deviation from sample size", and how do I calculate it?

I'm doing a research on 538's Pollster ranking and their methodology, and trying to replicate it on another database. However in the process of doing that, I couldn't find the 'expected error', error ...
Eunhyuk's user avatar
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Joint probability table not adding up to 1

I got this question asked in my exam today: Suppose there is a group of 9 students that belong to a university. 4 of them study economics, 3 of them study business and the rest study engineer. Suppose ...
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Get Conditional Density from Joint Density in Abstract Setting

Let $X_1$ and $X_2$ be real random variables defined on a common probability space $(\Omega, F, P)$. Let $\nu$ be a $\sigma$-finite measure on $\mathbb{R}^2$. We suppose that $X=(X_1,X_2)$ has joint ...
温泽海's user avatar
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Linear regression: $\hat{u}$ vs. $u$

I am studying statistics as a major and got some confusion watching $\hat{u}$ and $u$. My professor said $\hat{u}$ is called a residual and $u$ is an unobservable error term. Therefore this is what I ...
squid__'s user avatar
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Expected value of the random variable: $\int (\mu + \sigma Z)\phi(x)dZ$

There is a random variable $X$ and its standard normal viariable $Z = \dfrac{X-\mu}{\sigma}$. I'm looking for an expected value of the random walk defined by $V = \mu + \sigma Z$. For $X \in [0, \...
bag_dush's user avatar
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1 answer
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Confidence Interval for Expected Value of Binomial Distribution

been trying to wrap my head around this for a while. What is the confidence interval for the expected value for a binomial distribution? Let's say for a sample, I throw a coin 7 times and only 1 is ...
joelleoqiyi's user avatar
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49 views

Unbiased Cumulant Estimate - Fifth Cumulant

I am searching the definition of the $5^{th}$ unbiased cumulant estimate. Let $K_j$, be the $j$-th unbiased cumulant estimate of a probability distribution, based on the sample moments. Let $m_j$ be ...
claudioclaudio's user avatar
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1 answer
77 views

How to prove the inequality for the standard deviation of a linear combination of two random variables

The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book provides a conclusion about the risk and return of ...
bokabokaboka's user avatar
4 votes
4 answers
301 views

Probability that at least 1 student is taking a language class solution

This is a problem from A First Course in Probability by Ross: An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The classes are open to any of the ...
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Using Lebesgue Dominating Convergence Theorem to prove that T-distribution converges to Normal distribution as n goest to infinity. [closed]

I am trying to prove that the T-distribution converges to Normal distribuiton. In the process I need to use the LDCT. I understand that the LDCT makes it possible to interchange the limit and integral....
Hyewon's user avatar
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Does the kurtosis need to be finite for the sample variance to be consistent?

It is known (see this answer) that if $\mu_4$ is the fourth central moment of a distribution and $\sigma$ is the standard deviation, then we can write $$\operatorname{Var}(S^2_n)=\frac{1}{n}\left[\...
harrydiv321's user avatar
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1 answer
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Analysis of Hypersonic Missile Interception Rates by Cruisers

I've been exploring the mathematical concept of artillery, particularly focusing on the challenges posed by modern hypersonic missiles to interception rates of naval cruisers. My problem: DF-ZF ...
Köh's user avatar
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1 answer
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MGF for Unif(0,Gamma(2,\theta)

Let $X\sim Gam(2,\theta)$ and $Y\sim Uniform(0,x) $ Compute the pdf of $X|Y=y $ for $y>0 $ and $MGF_{XY}(t_1,t_2)$ $f_X(x)=x\cdot\frac{e^{-x/\theta}}{2\cdot \theta^2}$ $f_Y(y)=1/(x-0)=\frac{2\cdot ...
Hrackadont's user avatar
0 votes
1 answer
26 views

Compute inverse of a special 2 by 2 block matrix.

Let $$X\in\mathbb{R}^p,\quad \tilde{X} = (1, X^{\top})^{\top}\in\mathbb{R}^{p+1},\quad \tilde{\Sigma}=\mathbb{E} \left[\tilde{X} \tilde{X}^{\top}\right]\in\mathbb{R}^{(p+1)\times (p+1)} $$ $$ (\tilde{...
maskeran's user avatar
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4 votes
1 answer
97 views
+50

Dudley's inequality: Sending $\delta$ to $0$

Let $\{X_t\}_{t \in T}$ denote a mean-zero separable sub-Gaussian stochastic process with variance proxy $\sigma^2 = 1$, defined on some metric space $T$ with metric $d$. Dudley's integral inequality, ...
rubikscube09's user avatar
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-4 votes
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Can someone explain these Poisson Process Questions? [closed]

Let {N(t),t>=0} be a Poisson process with rate lambda. Suppose 0<s<t. Find A) E[N(t)|N(s) =5] B) P(N(s)=k|N(t)=8) C) E[N(s)|N(t) =8]
Alwin C's user avatar
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1 answer
24 views

Is there any way to compact the propagation of uncertainty formula in terms of vectors?

The uncertainty associated to $\xi=f(\mathbf x)$ with $\mathbf x\in\mathbf R^n$ is $$\delta\xi=\sqrt{\sum_{i=1}^n\left(\frac{\partial \xi}{\partial x_i}\delta x_i\right)^2}$$ What I was wondering was ...
Joan S. Guillamet F.'s user avatar
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1 answer
74 views
+250

Constructing most powerful test

Suppose we have a random sample of size $n = 1$ from the probability density function: $$f(x \mid \theta) = \begin{cases} 1 + \theta^2(0.5-x), & 0<x<1\\ 0, & \text{otherwise} \end{cases}$...
adisnjo's user avatar
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0 answers
44 views

UMVUE for $\rho$ when $X_i \sim N\left( 0,\sigma^{2} \right)$ and $Corr\left( X_{i},X_{j} \right)=\rho$ [closed]

Given $X=\left( X_{1},\cdots, X_{n}\right)$ and $X\sim N\left( 0,\Sigma \right)$, where $$ \Sigma=\sigma^{2} \begin{pmatrix} 1&\rho&\cdots &\rho\\ \rho &1&\cdots &\rho\\ \vdots ...
Chen Samuel's user avatar
1 vote
1 answer
43 views

UMVUE of $\mathbb{E}[X^2]=\lambda^2 + \lambda$ where $X\sim\mathrm{Pois}(\lambda)$.

This is the same question as this: UMVUE of $E[X^2]$ where $X_i$ is Poisson $(\lambda)$. Here, I restate the problem for completeness: Let $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} \mathrm{Pois}...
pbb's user avatar
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2 votes
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Deriving the CAPM pricing kernel from the general SDF and consumption-based kernel

I'm reading the paper "Quality minus junk" by Asness et al. (2019) and trying to understand the pricing kernel definition they provide on page 6. The authors present the following pricing ...
Newbie's user avatar
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Calculating the expected value for a complicated pdf

I am given the following cdf (cumulative distribution function) for the random variable $T$ with support $x\in(0, \theta)$. $$F_{T}(x)=\Bigg(\frac{x-\frac{1}{n}}{\theta}\Bigg)^{n}$$ I am asked to ...
HornyPigeon54's user avatar
-3 votes
0 answers
19 views

Stalker Prob Stats [duplicate]

There’s a new family with two children in the neighbourhood, since you’re a stalker and a statistician, you have found out that at least one of the children is a boy. a) Find the probability that both ...
Jee Aspirant's user avatar
0 votes
1 answer
38 views

Are sample mean and variance unique estimators?

I get that the sample mean and sample variance are unbiased and consistent estimators. But i'm wondering if they are unique estimators that are unbiased and consistent. i.e. for any statistic A, if A ...
Logan Lee's user avatar
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1 answer
20 views

Tricky Bar Graph or Histogram [closed]

I am trying to determine if this is a bar graph or a histogram. It's the distribution of American families by income. This has been such a struggle for me, but I'm sure there's a simple explanation. ...
spraka's user avatar
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1 vote
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+50

Exponential Family with Complete Sufficient Statistic

Suppose that $X$ is in an exponential family taking values in $\sigma$-finite space $(\mathcal{X}, F_{\mathcal{X}}, \nu)$ probability density function $f_{\theta}(x)=h(x) \exp \{\eta(\theta)^T T(x)-\...
温泽海's user avatar
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Strongest Result on Existence of Minimal Sufficient Statistic

Let $X$ be a random variable taking values in a measurable space $(\mathcal{X}, F_{\mathcal{X}})$ whose distribution $P_{\theta}$ is chosen from a parametric family of probability measures $\mathcal{P}...
温泽海's user avatar
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Can I find the pdf of a multivariate random variable with one fixed term by taking the limit of the variance approaching 0?

The projected normal distribution is a probability distribution on the sphere $\mathbb{S}^{d-1}$ obtained by projecting a Gaussian random variable $x \sim \mathcal{N}(\mu, \Sigma)$ onto the sphere as $...
dherrera's user avatar
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-2 votes
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29 views

Minimum win rate needed to get even

Let’s say that there is a game that awards you 1 star when you win a match and deducts 1 star when you lose. If you win 3 times consecutively, the 3rd win and subsequent wins will award you 2 stars ...
Aiden Chow's user avatar
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1 vote
1 answer
52 views

Why I am getting two different medians

let's say we have: $10, 12, 15, 10, 10, 12$ sorted: $10,10,10,12,12,15$ this way is clear that the median is $\frac{10+12}{2} = 11$ however, I was told that I could find the median using this table ...
samsamradas's user avatar
0 votes
1 answer
34 views

Hypothesis Test using Confidence Intervals

In $1965$, a newspaper carried a story about a high school student who reported getting $9207$ heads and $8743$ tails in $17950$ coin tosses. Is this a significant discrepancy from the null hypothesis ...
adisnjo's user avatar
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1 answer
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Law of Total Probability with Extra Conditioning Examples?

I cannot understand the logic behind the Total Law of Probability with Extra Conditioning. I am hoping to get some simple examples that explains it. The ones I have seen so far are still very ...
Max's user avatar
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23 views

Fisher-Rao distance and Hellinger distance

I have read on a tweet that "The Fisher-Rao distance is the geodesic distance associated to smooth divergences (eg. Kulback-Leibler). Without constraint, it is the Hellinger distance" and I ...
Ramufasa's user avatar
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36 views

If 8 new teachers are to be divided among 4 schools, how many divisions are possible? Each teacher is distinguishable? [closed]

I know that we must use the Multinomial Theorem. But why is $$n_1=n_2=n_3=n_4=1?$$ Is it because each teacher is distinguishable and can only be counted once? What does each $$n_1, n_2, ... ,n_r$$ ...
Iris Gu's user avatar

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