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

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iff $E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$

Question: $X_n$'s are i.i.d then $$E\Bigl(\sup_{n\geq 1} \frac{|X_n|}{n}\Bigr)<\infty \iff E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$$ My attempt: for $\Rightarrow$ part, because $\limsup ...
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38 views

Existence of complete sufficient statistics

Suppose $X_1,\ldots,X_n$ are iid r.v.'s, each with pdf $f_{\theta}(x)=\frac{1}{\theta}I\{\theta<x<2\theta\}$. I find the minimal sufficient statistics $(X_{(1)},X_{(n)})$. I am trying to prove ...
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33 views

Distance between two p.m.fs

I am stuck with the following problem from research. Is there any existing distance measure which can compare two probability mass functions with different support? For eg. for pmfs $p_1$ and $p_2$ ...
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1answer
39 views

What is the probability that statistical significance will be achieved in a second test?

Assume I'm thinking of investing in a small biotech company. In a phase 2 study with 120 patients split 80/40 between the new drug and old drug, the progression free survival (PFS) rate of the ...
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35 views

minimum kullback leibler estimator

Suppose that one has independent and identically distributed samples $x_i,i=1,...,n$ from some unknown density and one wants to fit a probability distribution $f_\theta(x)$, where $\theta$ is a ...
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1answer
36 views

Comparing Weighted Coins

Suppose we flip two different weighted coins each $n$ times. The first coin comes up heads $a$ times, the second coin $b$ times. When can we reject the null hypothesis that the coins are weighted the ...
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10 views

How to estimate the occurrence time of an event for scheduling purposes precisely?

I working on a system in which a number of requests arrive in random time. Consider the following timeline: In the above example, we have a certain number of components: ...
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21 views

Having difficulty with finding correct answer for binomial/probability question

I'm having a little difficulty with an assignment that I was just assigned, and I'm not too sure if I did it right but if anyone would like to check my answer to see if it's correct or not that would ...
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10 views

How to analyse parameter estimation results for large ODE model?

I am a systems biologist building ODE models of biological processes. I am using software that implements global optimization algorithms to identify the unknown parameters. Because I do not have a lot ...
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29 views

How can dimension reduction lead to better results?

Can someone please explain why a model fitted using a linear combination of the parameters can have better results (lower error) than a plain vanilla one with all the parameters? Can I think about ...
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27 views

Order Statistics for IIND Variables

Given 3 exponential random variables with different means (for example 1, 2, 3), how can one calculate E(X) for MIN(X1,X2,X3)?
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20 views

Bayesian Optimization and Averaging

I read in a statistics book that optimizing the likelihood function (or more generally Quasi-likelihood function) in a Bayesian framework is the same as averaging the posterior means. Why is this ...
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20 views

Classifying each variable

How can classify variables(number of the company, months, sector and size). I Thought... company = discrete quantitative variable months (watch below)= sector = nominal qualitative variable ...
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30 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of ...
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1answer
54 views

odds of all 15 MLB teams winning at home

The recent rare case of all $15$ MLB teams winning at home was reported as odds of $1$ out of $32{,}768$. I thought the calculation would be $0.5 \times 0.5 \times 0.5 \times \cdots$ for the $15$ ...
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34 views

Choosing something 35% more often

If I want to select an element from an array at random, the following will choose each element roughly at 1/L percent of the time: ...
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45 views

Stat: Probability to have one element of a combination identical to one element of another combination

For a business application, I currently have to provide the probability we are going to have an issue in one application. The combination is composed of N unique ...
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13 views

Universal polynomial approximation algorythm

I would like to ask, is there any universal algorythm to fill this matrix for any n value? $\textbf{A} = \matrix{n & \sum x_i & \sum x_i^2 & \cdots & \sum x_i^n \cr \sum ...
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26 views

Linear combination of i.i.d random variables

We say that a random variable $X$ satisfies the $(\alpha,\beta)-$condition for some $\alpha>0$ and $\beta>0$ if $$\mathbb{P}(|X|<t)<\alpha t\text{ and }\mathbb{P}(|X|>t)<e^{-\beta ...
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36 views

Probabilites, Statistics and Counting Questions

I'm practicing some GRE math problems which I don't have answers for. Would you please correct my answers if they're wrong? Q1. You have 4 books, A and B and C and D. In how many ways you can ...
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1answer
18 views

Sampling Gaussian Distribution, how to understand the formula?

To generate a sample according to Gaussian distribution, it follows an algorithms such that $\frac{1}{2}\sum_\limits{1}^{12}rand(-b, b)$, where rand function is to generate a sample uniformly on ...
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1answer
28 views

Rate of Return / Standard Deviation / Correlation Coefficient - Mathematical Finance

Consider these two stocks: AT&T Inc. (T) and Verizon Communications Inc. (VZ). Use the daily adjusted closing prices from March 1, 2015 to August 12, 2015 as historical data. Estimate the mean ...
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31 views

Integrate a Dirichlet Posterior over a finite interval

How do you integrate (analytically or numerically) a Dirichlet Posterior over a finite interval or union of finite intervals of events/rvs $x_i$
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31 views

Positive semi-definite in Linear model

Suppose $Y_{n \times 1} \sim N(X\beta,\sigma^2V)$ where $V_{n\times n}$ is invertible and $X_{n\times p}$ is of rank $p$ and $\beta_{p \times 1}$ is unknown and to be estimated by $Y$ and $X$. Which ...
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1answer
11 views

T distribution problem

I will be using $t$-distribution to solve this problem. Specifically,the pooled variance test because both samples have size less than $30$,and both populations seem to have the same population ...
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2answers
21 views

How to calculate probability with Z score not on table?

According to this Z table in my book, anything with a $z$ of over $3.16$ is probability 1, but this is not right. The textbook also has an example where $\mathbb P(Z\geq3.9) = 0.000048$; can somone ...
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1answer
19 views

Determining the MVUE of $ f(x;\theta) = \theta^x (1-\theta)$.

The Statement of the Problem: Let $X_1, X_2, ... , X_n$ be a random sample from $$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$ (a) Find the ML estimator of $\theta$. (b) Show that $T ...
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50 views

Bounds of integration for PDF

Given the PDF $\ f_{XY}(x,y)=8xy $ $\ 0<x<1 $ $\ 0<y<x $ Find $\ f_Z(z) $ Where $\ Z=\frac{Y}{X}$ I want to calculate $\ F_Z(z) $ so that I can derive it and find $\ f_Z(z) $. ...
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1answer
30 views

Descriptive statistical analysis

I have some data obtained from questionnaire. The question asks to perform a descriptive statistical analysis for the above data and hence interpret your results. What does it mean? Is it mean I have ...
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18 views

Overlap between disks generated through a Poisson process

I have a homogeneous spatial Poisson process on $[0-d; 0-d]$ with an expected number of points equal to $n$. I use the points to generate disks of radius $r$ without any overlap constraints. I am ...
3
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1answer
26 views

Is the chance of a variable also a parameter for a probability distribution?

I'm new to statistics and I'm a bit confused about the concepts of 'chance of a variable' and 'parameters of a probability distribition'. Is chance also a parameter? And if so: can computing the ...
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5 views

Error in Joint Modelling in R

I have a dataset which has the following variables: sex (indicator), risk (indicator), and age. I've done cox models with these variables no problem and even basic joint models with them no problem. ...
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1answer
19 views

Estimate of mean weight?

An Argentinean Rancher, who raises miniature Llamas, wants to have an estimate of the mean weight of his herd. The herd could be regarded as being a Normal population, with variance 225 lbs. He ...
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2answers
21 views

What kind of permutation/combination is this?

I have 2 letters to choose from, and 3 positions I can place them in, where order matters. Lets say letters A and B: The possibilities, from an intuitive sense are: AAA, BAA, BBA, BAB, ABB, BBB, ...
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2answers
35 views

a problem regarding conditional probability and binomial distribution.

Die A has 4 red and 2 white faces whereas die B has 2 red and 4 white faces . A coin (fair) is tossed once . If it falls head , the game is carried on by throwing die A alone. If it falls tail die B ...
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If $\theta = (59.3\pm 1.2)^{\circ} $, find $\tan(\theta)$ and it´s uncertainty.

How can I find the uncertainty? I need to use this formula: The general formula for uncertanties is: $\sigma_q = \sigma_q (a,b,c,...)$ and $\mathrm{Cov}(a,b,c,...)=0$ $\,\,$then $$\sigma_q ^2 = ...
3
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1answer
38 views

How to quantify how much my data resembles a linear relationship?

I have a bunch of data points in Excel for different test subjects that are each represented by unique colors in the following graph: I wish to quantify in Excel what seems obvious to my eyes, that ...
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1answer
66 views

Questions on finding expected value and variance on a Poisson distribution

FProblem: A student walks along a real line and tries to get to the origin. Each step he makes is random ; the larger the intended step, the greater the variance is of that step. When the student is ...
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1answer
54 views

Question about normal approximation on roulette

Problem: At roulette, you bet a dollar on red $30$ times in a row. Each time you win a dollar with prob. $18/38$ and lose a dollar with prob. $20/38$. Find approximately the probability that after ...
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2answers
37 views

Conditional variance of sum of two correlated random variables

Let $\theta\sim\mathcal{N}(\bar{\theta},1/\tau_\theta)$ be a normally distributed random variable, $\varepsilon\sim\mathcal{N}(0,1/\tau_\varepsilon)$ be a normally distributed noise term independent ...
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44 views

Question about normal approximation

Question: A class has $20$ students, each student tosses a fair coin $300$ times and records the number of heads. Approximately what is the probability that no one gets exactly $150$ heads? (Use ...
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0answers
19 views

Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$. Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: ...
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1answer
19 views

Multidimensional Central Limit Theorem

Let $\mathbf{U}^n = (U_1^n,U_2^n,\cdots, U_k^n) $ converge in distribution to $N(\mathbf{0},\Sigma)$. My question is that if $V_{k+1} \sim N(0,1)$ and is independent of $\mathbf{U}^n$ then is it true ...
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0answers
13 views

Autoregressive process and average return time

I am back in my maths courses but I doesn't arrive to assess the average return time of an $\operatorname{AR}(1)$ process. Has someone any idea? And after for an $\operatorname{AR}(2)$? Suppose ...
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1answer
17 views

A question about Fisher–Neyman factorization theorem

$f_{\theta}(x)$, then $T$ is sufficient for $\theta$ if and only if nonnegative functions $g$ and $h$ can be found such that $f_{\theta} = h(x)g_{\theta}(T(x)) $ The statement is: if $F(t)$ is a ...
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1answer
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Determining the efficiency of $2 \overline X$.

Let $X$ be a random variable with pdf $$ f\left(x;\theta\right) = \left\{ \begin{array}{lr} \frac{3\theta^3}{(x+\theta)^4} & \text{if } 0<x<\infty \text{ and } 0 < \theta ...
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39 views

Evaluate forecasting quality

A statistical model calculate the probability that some independent event occurs. These events either occur or not (binary outcome). I am looking to evaluate the quality of my forecasting. I looked ...
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26 views

Avoiding over/under fitting & validation set size

I've already posted this question on stats.stackexchange a couple days ago, maybe someone here can help. I'm using a PLS model associated to FT-NIR spectroscopy to predict some properties of ...
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2answers
51 views

Finding a transform of X

Question: Y~ Uniform (0,1) and given Y=y, random variable X is binomial with parameters n and y. Find the transform (moment generating function) of X. Does X have a special name? Attempt: I was ...
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2answers
26 views

the meaning of Confidence Interval

I am a bit confused how we interpret confidence intervals. For the formula $\bar{X}\pm z_{\large\frac{\alpha}{2}}\large\frac{\sigma}{\sqrt{n}}$ ,we substitute $\bar{x}_{obs}$ for $\bar{X}$, but ...