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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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How to turn a non-stationary AR process into a stationary process? (ARMA modelling)

In my notes it says for non-stationary processes, forecast differences. E.G. For, y(t)= 1.5y(t-1)-0.5y(t-2)+ε(t) *: which is non-stationary as the coefficients sum to 1 Forecast, Δy(t) = 0.5*Δy(t-...
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Computational Statistics: Variance reduction

In this exercise, I should do the following subquestions for $\theta=P(X+Y\geq 15)$ where $X\sim Exp(0.5)$ and $Y\sim Exp(0.25)$ Implement the following estimators for $\theta$: (i) The raw ...
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Given two unbiased estimators, determine the optimal k* s.t. the corresponding θk* is the most efficient

Consider θa and θb are uncorrelated, unbiased estimators of θ and let θa(k)+θb(1−k)=θk. Determine the optimal k* s.t. the corresponding θk* is the most efficient (suppose MSE(θa) and MSE(θb) are σ^2a ...
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20 views

population standard versus sample size

I was wondering what is the difference between the terms of "population standard of 1%" and "sample size of 1%" for the assessment of uniformity. Thanks
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3answers
35 views

Find distribution of $Z=\frac{X+Y}{2}$ given $f_{X,Y}(x,y)=e^{-(x+y)}$

Excercise Let $X, Y$ be random variables such that their joint density function is defined by: $f_{X,Y}(x,y)=e^{-(x+y)}, \enspace x,y>0$. Find the distribution of Z defined as: $Z=\frac{X+Y}{2}...
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2answers
24 views

Given the digits of a passcode, what is the probability of getting 4 in their correct location?

Say you have a passcode $135790$ and a burglar knows exactly what digits it is made up of and that there no duplicates, but not the correct ordering. How do you calculate the probability that he gets ...
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11 views

How to find best achievable significance level?

Given a certain $n$ number of samples, the null hypothesis states that at most 50% of those participants lean towards one answer on a survey, which is rejected only if the sample evidence strongly ...
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16 views

Bias of the method of moments estimator in a Gamma distribution.

Given a Gamma distribution $$ g(x;\alpha,v)=\frac{\alpha^v}{\Gamma(v)}x^{v-1}e^{-\alpha x} $$ I simulated this gamma distribution with $\alpha=0.2$ and $v=1$ with the software r $100$ times. I want ...
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15 views

How D4 is calculated (Range Charts)

The formulas for the chart are found below: https://stats.stackexchange.com/questions/185373/closed-formula-for-d4-constant-calculation-moving-range-chart-constant My question is what the x1, xn ...
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1answer
30 views

Hypothesis Test when unknown mean and unknown variance

let have a random variable X which follows a normal distribution of mean $\mu$ and variance $\sigma$. we want to carry out the following hypothesis test: $H_0: \mu=\mu_0$ against $H_1: \mu\neq\mu_0$ ...
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Die roll and hypothesis testing

First of all this is not a homework or an assignment; it's an exam question that I couldn't solve. "There is a $4$-sided die that has the same probability of showing up for each face when rolled, ...
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13 views

central region of Normal distribution and Exp distribution

This is the question in my homework i do not know how to solve it enter image description here
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1answer
26 views

The expected value of a single observation

I'm struggling to understand a line of the following proof: Claim: The sample mean $\bar{X}$ is an unbiased estimator of the population mean $\mu$ Proof The random variables each based on a single ...
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1answer
16 views

Problem: How many 'timetables' can be created for visiting 8 different lectures without time clashes?

Apologies for the poorly worded question but it really requires more explanation than can be done with one line. I have a statistics problem, or rather, a timetabling problem that I feel can be solved ...
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13 views

Sketching residual plots

A set of data has been analysed by fitting the model $Y_i = x_i^{'}\beta + \epsilon_i$, where $\epsilon_i$ follows a normal distribution with mean 0 and variance $\sigma^2$. Sketch a residual plot for ...
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8 views

the moment generating function of the negative binomial distribution

According to the textbook I use, it states that: $X$~$Neg.bin(x;k,\theta) = {n-1 \choose k-1}\theta^k(1-\theta)^{n-k}$ Which I have no problem. The problem arises when I try to find the moment ...
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1answer
32 views

Probability of a random subset of Z

I'm stuck in this question, could someone give me a hand? I'll post what I've done so far. Question 9: Let $A=(1,2,3,4)$ and $Z=(1,2,3,4,5,6,7,8,9,10)$, if a subset B of Z is selected by chance ...
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0answers
23 views

Consistency of maximum likelihood estimator for negative binomial distribution

Let $X = (X_1,\ldots, X_n)$, where $X_1,\ldots,X_n$ are independent and have the same distribution: $$P_{\theta}(X_i = k) = \binom{k + r - 1}{k} \cdot \theta^r \cdot (1 - \theta)^k~,$$ with $k \in \...
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1answer
10 views

factorial moment generating function

I'm trying to get the factorial moment-generating function of a binomial random variable. I know that $F_X(t) = E[t^x] = \Sigma_xt^xp(x)$ so I get $\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$ ...
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1answer
13 views

find the density function of $(U,V)$

Let $X=(X_1,X_2)^T$ be a random vector with 2-dimensional normal distribution, $E(X_1)=E(X_2)=0 , Var(X_1)=Var(X_2)=1$ and $Cov(X_1, X_2)= \nu$ with $|\nu| <1$. And let $Z \sim Bin(1,\alpha)$ be ...
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1answer
18 views

Predicting capacity problems based on historical data

Situtation: An oil-company collects waste oil from small car/truck workshops. The workshops have only limited waste oil storage varying per workshop. Oil must be collected from those workshops when ...
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2answers
39 views

If $X_n$ and $Y_n$ are independent does $(X_n,Y_n)\overset{d}{\rightarrow}(X,Y)$?

More formally: If $X_n\overset{d}{\rightarrow}X$ and $Y_n\overset{d}{\rightarrow}Y$ and also $X_i$ and $Y_j$ are independent for all i,j; does $(X_n,Y_n)\overset{d}{\rightarrow}(X,Y)$? I am aware of ...
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0answers
8 views

False-positive in a mixture distribution

This is an example of a photoluminescence intensity signal. Suppose you have a mixture of two normal distributions for the signal data. One normal distribution is due to negative results while the ...
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0answers
13 views

To code for principal component in functional PCA [on hold]

I am trying to get the way to find eigenfunction from eigenvalues. As far as I understand from reading that I have to multiply eigenvalues by something to get principle components in functional PCA. ...
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0answers
12 views

Differential Corrections

Can you explain to me this please? How is $\chi^2$ computed with this? How is the accuracy improved by replacing $\Delta p_k$ by $\Delta p_k/2$? What is the result? $p_k$ was at the biggining, so its ...
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1answer
34 views

Finding the conditional expectation of independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find (a) $E(MX|M=X)$ (b) $E(MX|M=Y)$ (c) Cov$(X,M)$ (a) I first ...
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0answers
13 views

Estimated value of sample variance for normal

Let $X_1 \ldots X_n$ be iid $N(0, \sigma^2)$ observations. I'm told that the correct answer is $E[S^2] = \sigma^2$. If I use the given value of mean in my sample variance calculation then I get the ...
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0answers
19 views

How do you create the polynomial basis function given data?

According to Wikipedia: However, i saw that basis functions applied before linear regression looks like: Is this saying that, Each observation x has D dimensions. then applying the wikipedia ...
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0answers
14 views

Using measurement scales with sets of data

When considering the four scales of measurement (nominal, ordinal, interval, and ratio), I'm wondering whether they are applied only to objects or data points, or whether they can also be applied to ...
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26 views

UMVUE of Geometric Distribution [on hold]

What is the UMVUE of $p^2$ for geometric distribution?
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0answers
19 views

Something like the Wilson confidence interval, but taking into account the “typical” success rate?

If I understand the Wilson confidence interval correctly (and I'm not confident that I do), it doesn't seem to take any data into account other than the success/failure counts of the particular thing ...
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31 views

How to divide data in half using linear line?

Say we have data: $(x,y) = (1,20) (2,9) (3,18) (4,15) (5,14) (6,9) (7, 4) (8,39) (9,25) (10,11)$ How would one create a slope formula that divides equally so that half the data points are above (50%...
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0answers
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Angle between regression line and SD line when plots flipped

The question I am working on is: The scatterplot y* vs x* has a regression line that makes an angle with SD line. Then the corresponding angle for the scatterplot of x* vs y* is: I have generated ...
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0answers
17 views

Estimating population size of endangered parrots

I am trying to find the best way to analyse the results of a study I have just coordinated. In short, I want to be able to estimate the total population size of a species of parrot based on ...
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0answers
15 views

Best linear prediction as a projection in a Hilbert space $L^2$

Consider two random variables $Y$ and $X$. In the context of the best linear prediction, if we would like to predict $Y$ given $X$ known, we derive the solution solving the following minimize problem ...
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0answers
10 views

Effective way to calculate Average and Median of a historic series

I am working with a problem, where in I store the feature usages of any user on a daily bases. After each day the data gets accumulated and I wanted summarize the data and generate pre-calculated ...
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0answers
5 views

Algorithm for producing a Moving Average (MA(q) as in ARIMA) model.

I have a time series $X_t$ and I want to produce an ARMA forecast (without using any automated packages - the purpose of my project is to understand how those work). So far, I have the AR(p) part ...
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0answers
17 views

How to aggregate the normal distributions of two Kalman populations?

Suppose I have the following Bayesian Network: It's given by the following probability distributions: $$\begin{aligned}X_1&\sim \mathcal N(\mu, \delta^2)\\ \forall i, 2\leq k\leq n: X_i|X_{i-1}&...
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1answer
23 views

Optimising unrounding using maximum likelihood

I have a bunch of rounded random independent numbers. I want to replace them with unrounded numbers such that the unrounded numbers are 'most likely' to have been generated by some (continuous) ...
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0answers
23 views

Fitting the data to the distribution

I have been given data as hw, where each have $10000$ entries for a loss of $x$ claim and loss of $y$ claims. I have to find the best distribution out of $4$ options to match the data, I have ...
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1answer
15 views

How to show that the multivariate normal depends on the data only through $\Sigma x_n$ and $\Sigma x_n x^T$

I've shown something similar for the 1-dimensional case. That $\Sigma x_n$ is the sufficient statistic of the gaussian mean and $\Sigma x_n$,$\Sigma x_n^2$ are the sufficient statistics of the ...
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1answer
13 views

What if correlation coefficient is much less whereas the p-values is much lesser than the our significance level?

I have a data between rating of the restaurant and the approximate price of food in the restaurant. After doing Hypothesis testing. The pearson correlation coefficient was found to be 0.33 whereas p ...
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0answers
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How to reproduce this influence statistic found in this Taguchi method example?

Objective: to recreate the influence values given in this online tutorial. The context for this question is from [Design of Experiments][1] from Taguchi. Thank ...
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2answers
39 views

Standard Normal Distribution - the mean profit

The life of a certain type of car battery is known to be normally distributed with mean $30$ months and standard deviation $6$ months. The batteries cost the manufacturer $\$40$ each to make, and ...
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1answer
18 views

Tail bound for sum of random variables satisfying subgaussian upper tail bound

So suppose you have a collection of random variables $X_1, \cdots X_n$ that are iid and they all satisfy the tail bound $$ P(X_i-L>u)\leq \exp(-\frac{u^2}{2\sigma^2})$$ for all $u>0$. Is it ...
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1answer
12 views

Sampling with and without replacement.

Prove that the probability of drawing a unit at any draw from a population of size N, remains same in without and with replacement sampling scheme. I know how to prove that this probability is $\frac{...
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0answers
31 views

Correct interpretation of confidence interval

I think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained. My ...
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0answers
33 views

How to convert vectors of unequal length to the same size [on hold]

I have a set of time series vectors that have varying lengths and I am trying to determine a method using which all of them could have the same specified length.
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0answers
20 views

Find $F ^{−1} (u)$, for $u ∈ (0, 1)$ by solving the equation $u = F(x)$ for $x$

I am working through a problem. I began with the pdf: $f(x)=\frac{1}{2}\alpha*e^{-\alpha|x|}$, where $\alpha > 0$ and $x ∈ (−∞,∞)$. I found the cdf to be: $$F(x) = \begin{cases} \frac{1}{2}e^{\...
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1answer
18 views

What is the covariance matrix of $4\times2$ data points? [on hold]

I do not know how to calculate the covariance matrix for a $4\times2$ data points The points are as followed: X | Y -2 | -1 -1 | 0 1 | 0 2 | 1 Can I get ...