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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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Question on definition of unbiased estimator

The definition in my textbook is the following. Let $\big(\Omega, \mathcal{F}, (P_\vartheta:\vartheta\in\Theta)\big)$ be a statistical model and $(E,\mathcal{E})$ be a measurable space. If $E$ is a ...
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how write residual by adding outliers?

I'm reading a paper about outliers and its detection. there they suppose a initial group of observatios and next they add a group of high leverage identical outliers. where to the initial observations ...
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24 views

Find probability mass function of $Y=N-X$

Let $N$ and $X$ distributed Poisson random variables with respect parameter $\lambda$ and $\lambda\cdot p$. Let $X\vert N$ have binomial distribution with number of trials $n$ and success probability $...
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1answer
15 views

Finding the log likelihood of poisson and normal models that have a log link function.

I have these two distributions, $$ \begin{split} Y_i &∼ \mathrm{Pois}(λ_i)\\ \log(λ_i) &= β_0 + β_1x_i \end{split} $$ and $$ \begin{split} Y_i &∼ N(µ_i, σ^2)\\ \log(µ_i) &= γ_0 + ...
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22 views

proof related to gamma distribution function [on hold]

finding gamma distribution from the given probability distribution function as shown in this picture A random variable x is said to be gamma-distributed with index α if its probability density ...
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1answer
21 views

Calculating the Cramer-Rao lower bound for the variance of unbiased estimators of $θ$

Assume the standard situation, that is, let X1, . . . , Xn be independent and identically distributed with $X_k ∼ P_θ$, where$P_θ(x) = 1/(2θ^3).x^2.exp(−x/θ) , 0 < x < ∞$. This is a special case ...
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1answer
29 views

Find $c \geq 0$ so that $c\hat{\vartheta}$ is unbiased

I have found the following statistical model. Say $\Omega:=[0,\infty)^{n}$ and $P_{m}$~$[\mathcal{U}(0,m)]^{\otimes n}$ where $m$ is the parameter $\in [0,\infty)$ and $n \in \mathbb N$ Define the ...
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1answer
41 views

Sampling distribution. How large does the sample size need to be?

I am building a data analysis model for clinics visits. Basically, I want to capture the patient arrival distribution for each specific visit type at each specific time. Let's say the previous data ...
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1answer
45 views

Maximum Likelihood Estimator and Cumulative distribution for an i.i.d distribution $P_θ(x; θ)$

Assume the standard situation, that is, let $X_1, . . . , X_n$ be independent and identically distributed with $X_k ∼ Pθ(x; θ)$ , where $P_θ(x; θ) = 2x/θ^2$ if $0 ≤ x ≤θ$ and $0$ otherwise It is ...
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Find the mean and variance of $\hat{θ}$ for a special case of Gamma Distribution.

I've completed most of the question, however i'm not sure if i'm correct so far in order to proceed with the rest of it. Find the mean and variance of $\hat{θ}$ for a special case of Gamma ...
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1answer
35 views

Probability mass function from a generating function

I have the generating function $G_x(\theta) = \frac{\alpha-1}{\alpha-\theta^2}$ and I am trying to determine the probability mass function. I believe I need to determine the Taylor series expansion ...
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Computation of marginal distribution for uncertainty quantification of dependent variables

In a few words, I have some dynamics with uncertainties in the initial conditions. I am using the Liouville equation and the method of characteristics to propagate in time the distribution of these ...
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2answers
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Formula to recalculate Variance after removing a value and adding another one given old variance

Let's say I have a data set of $10,20,30$. My mean and variance here are mean= $20$ and variance = $66.667$. Is there a formula that lets me calculate the new variance value if I was to remove $10$ ...
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1answer
27 views

how to bound the hat matrix?

I'm reading a paper about linear regression and in some point they define: $$ w_{ij}=\frac{h^{2}_{ji}}{ph_{ii}(1-h_{jj})^{2}}$$, where $h_{ij}$ are the elements of the hat matrix. The problem is ...
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116 views

The probability that a Binomial Distribution deviates from its mean by one standard deviation

Let $X$ be a random variable that follows the Binomial Distribution $\text{BIN}(n,p)$, where $n$ is a positive integer while $p\in(0,1)$. Its mean is $np$, and standard deviation is $\sqrt{np(1-p)}$. ...
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Real world retail statistical math problem

I'm currently working in a relatively large retail company and I've been assigned the task to measure the effect of a new assortment that is being tested in three trial stores. The idea is to compare ...
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Coordinate-wise gradient descent converges to least-squares solution

Does somebody know a reference (or maybe short proof/argument) for the following claim: Coordinate-wise gradient descent converges to a least-squares solution. Coordinate-wise gradient descent: ...
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40 views

Expected value of symmetric random variables

A random variable $X$ is symmetric if $X$ has the same probability distribution as $-X$. In the discrete case symmetry means that $P(X = k) = P(X = -k)$ for all possible values of $k$. In the ...
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25 views

Moment List of Exponential [on hold]

I need to follow this outline to find the moment list of Y ~ Exp(𝜆), 𝜆 = 1/E(Y) > 0. Let X ~ Exp(1). Show that X/𝜆 ~ Exp(𝜆), by finding the cdf of Y and then differentiating it. Write, L(t) = ∫∞ �...
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2answers
55 views

If the sample mean converges in distribution to F, the mean of the odd and even observations converge as well

could somebody please help with this question: Consider a random sample of $X_1, X_2, \ldots ,X_n$, such that $E[X_i] = \mu$ and $n $ is even. Define $P_n = \frac{2}{n}\sum_{i}X_{2i}$ and $I_n = \...
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2answers
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Bayes formula help!

Suppose a test for diagnosing a certain disease is successful in detecting the disease in $95$% of all persons infected, but it incorrectly diagnoses $4$% of all healthy people as having the serious ...
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32 views

Continuous random variables with zero variance

I have a very simple question. I heard in my class that a random variable is equal to zero if its variance is equal to zero. I understand that if the variance of a random variable is zero, then that ...
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1answer
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Is a study with a small sample size less meaningful than one with a big sample size?

To my understanding, the p value of a study determines how likely a result this extreme or more extreme would happen if there were no effect (e.g. due to coincidence, measuring inaccuracies etc.). ...
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Equivalence of two expected value terms and the distribution of its parameters

Problem: "The number of breakdowns $Y$ per day for a certain machine is a Poisson random variable with mean $\lambda$. The daily cost of repairing these breakdowns is given by $C=3Y^2$. If $Y_1,...,...
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1answer
78 views

What is the distribution of $(1+X^2)e^{-X^2/2}$ when $X$ is Cauchy?

If $X$ is a Cauchy random variable with $f(x)=\frac{1}{\pi}\frac{1}{1+x^{2}}$, what is the distribution of $Y= (1+X^2)e^{-X^2/2}$ ? What I tried: I was thinking I may be able to use Jacobian, but I ...
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1answer
13 views

Set theoretic notation for statistical sampling without replacement?

I would like to know how to use set theoretic notation to define a subset $D$ of a finite event space $S$ that contains exactly $n$ elements $d_i$ which are sampled uniformly without replacement from $...
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6 views

Logit link function for large $|\eta|$

I am doing a project on beta regression and have run into a problem with my link function $$g(\eta) = log[\frac{\mu}{(1-\mu)}]$$ $\eta = \sum x_i^T\beta$, with $x_i$ being vectors of explanatory ...
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1answer
33 views

Properties of stochastic ordering

On the entry for stochastic ordering in Wikipedia it is stated that If $A\preceq B$ and ${\displaystyle {\rm {E}}[A]={\rm {E}}[B]}$ then ${\displaystyle A{\overset {d}{=}}B}$ (the random variables ...
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Likelihood ratio criterion for testing equality of means in 2 normal populations

Let assume we have the following $$ X_{11}, X_{12},...,X_{1n_{1}} \sim N_{p}(\mathbf{\mu_{1}},\mathbf{\Sigma_{1}}) \\ X_{21}, X_{22},...,X_{2n_{2}} \sim N_{p}(\mathbf{\mu_{2}},\mathbf{\Sigma_{2}}) $$ ...
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1answer
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Is the use of GLS appropriate in case of statistically independent errors in linear regression?

Let $Y_i = \beta x_i + e_i $, where $e_1 ~ N(0, \sigma^2)$ and $e_2 ~ N(0, 2\sigma^2)$, and $e_1$ and $e_2$ are statistically independent. If $x_1 = 1$ and $x_2 = -1$ obtain the weighted least squares ...
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1answer
16 views

Degrees of freedom of the set of positive definitive matrices

If I am not wrong, the set of definite positive matrices with real coefficients is a convex cone without the vertex, which is the null matrix. What is the number of degrees of freedom for this set of ...
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20 views

Joint Bernoulli distribution given marginal distributions and correlation

General Info: I am in a situation where I have the marginal distributions for some number of Bernoulli random variables and a fixed correlation between each. I need to know the joint distribution of ...
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22 views

Lottery game with replacement

A state lottery game is played by selecting three numbers between 0 and 9, inclusive, with replacement. The winning numbers are selected in the same way. You win if your numbers match the winning ...
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1answer
14 views

Standard normal covariance

I am given that X is a standard normal distribution. Why is $Cov(X, -X) = -1$? I know that $Cov(X,X) = Var(X)$ and that the $Var(X) = 1$. Is the $Var(-X) = -1$?
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1answer
25 views

Statistics Independence Question: Given that an item has passed inspection, what is the probability that it is actually flawed?

The problem I have a question about is below. I only have a question on part (e), but I included the other parts of the question and answers as a reference. A quality control inspector is examining ...
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51 views

When is the covariance matrix $E\left(f(X)X^{\top}\right)$ invertible?

Let $X$ be a zero-mean absolutely continuous random vector in $\mathbb{R}^N$. Question: which class of continuous functions $f\,:\,\mathbb{R}^N\rightarrow \mathbb{R}^N$ yields an invertible matrix $E\...
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1answer
38 views

Lemma of Neyman Pearson (how to get critical region)

We have the Random Variable X, which is $\Gamma(p, \lambda)$ distributed with the density: $$f_{p,\lambda}(x)=\frac{\lambda^p}{\Gamma(p)}\cdot x^{p-1} \cdot e^{- \lambda x}$$ with $p = 10$ and $H_0:...
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3answers
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What is the probability of drawing two identical poker hands in a row from 1 52-card deck?

Same ranks, not same suit. Without replacement, 5 card hands, standard 52 card deck (no jokers). Example: draw A23QK draw A23QK (different suits)
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Calculating error on slope of graph

I'm trying to find the rate of change and the error on that rate based on 7 measurements points and the assumption that the trend is linear. My calculations are below: $$ \begin{array}{|c|c|c|c|} \...
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1answer
19 views

Indicator expected value

Let $X$~$Uc(0,1)$ You flip a coin twice with probability $X$ to get "Heads". Let $Y$ is the number of Heads you get . Let $I$ be an Indicator that get the value 1 if $Y=2$ and $0$ otherwise. What is ...
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43 views

Finding a sufficient statistic when density function is given

Let $\mathbf{X}=(X_1,X_2,...,X_n)^T$ be a simply sample of random variable $X$ whose distribution belongs to family $\mathcal{P}=\{ f(x; \lambda, \eta, \mu ), 0<\lambda, \eta <\infty, -\infty &...
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Takens theorem anomaly detection

Can takens theorem ourtperform the state of art for anomaly detection in multiple time series? It's really neural networks the solution of all or there are some new prespective in statistic for this ...
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22 views

What is the name of this Correlation matrix?

$ \left[ {\begin{array}{cc} 1 & p & p^{2} & ... & p^{n}\\ p & 1 & p & ... & p^{n-1}\\ p^{2}&p&1&...&p^{n-2}\\ ...\\ p^n&p^{n-1}&...&...
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1answer
24 views

Finding the condition on $k_1$ and $k_2$ of an unbiased estimator

I'm taking a statistics course and am asked the following : Suppose that $X$ and $Y$ are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the ...
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Method of Transformation in Statistics

I am trying to get a handle on finding pivotal quantities to use in confidence intervals. I came across a question regarding a uniform distribution: Suppose that we take a sample of size $n = 1$ ...
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28 views

Modeling distributed densities or is it?

Lets say that I am counting pixels over time. Each pixel is either the color red or blue. Let's say that I make up some threshold for the number of red pixels that I count and that when I reach this ...
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11 views

Robustness of Kolmogorov-Smirnov test

I was comparing the similarity of distribution of two data sets using KS test. Two data sets are not pure normal. I was wondering how much KS test depend on the distribution of samples. If the samples ...
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How find the Cook's distance expected value for a large n

I'm trying to prove that for a large n, the Cook expected value is approximated by: $$E[D_{i}]\approx \frac{h_{ii}}{p(1-h_{ii})}$$. Do you know where can i read the prove of that?. i tried to do it ...
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a lesson in descriptive statistics [on hold]

The average number of unemployed individual in a certain City in the past 5 years is 11,900 with a standard deviation of 1800. At least 65% of the average number of unemployed individuals will fall ...
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19 views

Normalize with respect to weight percent contribution

I am looking for some mathematical solution to calculate normalize score for an Index comprising multiple stock information. The Index has 5 companies stock information and each company has ...