# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### calculating CDF of kth order statistic

I have recently started probability and statistics on my own. Pls help in understanding below. $x_{(k)}$ is kth smallest random variable from sample of n iids ($x_1 \to x_n$) For calculating CDF of ...
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### Can the mean of a frequency distribution be regarded as

Can the mean of a frequency distribution be regarded as a weighted mean with relative frequencies as weights ?
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### Generate synthetic data given AUC for multivariate X

I need to simulate data for a fixed AUC and for multivariate X. I came across the link below which explains for the univariate x. Generate synthetic data given AUC. Has anyone had any ideas or ...
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### Joint distribution from multivariate Normal distribution

Consider two random variables $\{Y_1, Y_2\}$, which follow a joint multivariate normal distribution: $Y = [Y_1, Y_2]^T,$ \begin{equation} Y_1 \sim \mathcal{N}(\mu_1,\sigma_1),\; Y_2\sim\mathcal{N}(\...
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### Bayesian inference: all at once vs one at a time

Suppose that I have a prior on a parameter $\theta$ and update this prior in light of the realisation of $n$ random variables. It seems plausible that it is equivalent to update the prior $n$ times, ...
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### How to generate a random positive definite matrix in R of a given dimension [closed]

I am writing a function in which the input is a correlation matrix and I need to generate a random positive definite matrix of the same dimensions. I am unsure what the code is for this?
30 views

### Days as dummy variables

How should you treat "days" as a variable from a statistical perspective. The "days" variable can be defined as an integer describing the day of the year as follows: days = 1,2,3,4....
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### Who would you estimate the total user bade of a social network (Twitter, Faceboo,etc.)? [closed]

I need help with this, I don't know how to start, I don't remember any form for a estimator that estimates the total population
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### Why is data normalization important in Monte Carlo simulation? [closed]

I'm trying to understand why data normalization is important in a Monte Carlo simulation using an approximation equation - since that approach can predict arbitrary shapes and functions with random ...
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### Beta-Binomial conjugate proof

Can someone explain this proof to me? I get stuck on the transition from the third line to the last line. Namely: Is the integral being evaluated or not? How does the entire expression reduce to a ...
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### Prove that for any 2 events A and B , $P(A) + P(B) - 1 ≤ P(AB) ≤ P(A) ≤ P(A\cup B) ≤ P(A) + P(B)$

I want to prove 𝑃(𝐴∩𝐵)⩾𝑃(𝐴)+𝑃(𝐵)−1. How can I simplify the following proof? 𝑃(𝐴∩𝐵)=𝑃(𝐴)−𝑃(𝐴∩𝐵𝖼)=𝑃(𝐴)−1+𝑃(𝐴𝖼∪𝐵)=𝑃(𝐴)−1+𝑃(𝐴𝖼)+𝑃(𝐵)−𝑃(𝐴𝖼∩𝐵)=𝑃(𝐴)+𝑃(𝐵)+𝑃(𝐴𝖼∩𝐵𝖼)−1⩾�...
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### Most powerful test of size zero for $\theta$ given random sample from $U(0, \theta)$

I found a couple of questions on this site (Most powerful test of simple vs. simple in $\mathrm{Unif}[0, \theta]$ and UMP for $U(0,\theta)$ (simple x simple hypothesis)) that are similar to my problem,...
25 views

### Propose a code on R? [closed]

$1$-Objectives : We provide for every student a dataplay. i.e., a vector $(X_1;...;X_{500})\in\mathbb{R}^{500}$. This vector is the realization of a 500-sample $(X_1;...;X_{500})$ of a random variable ...
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### Exchangeability and joint distribution

The definition of an exchangeabilty for a finite sequence says that, if we have random variables $X_1,\ldots,X_n$, then for each permutation $\pi: \{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$, the joint ...
9 views

### even after multiple questions how is variance helpful over standard deviation in building a theory [closed]

How is variance as a concept preferred over standard deviation to build theory.? Can I have two scenarios with the same variance values
34 views

### Find distribution of the transformation $Y'(I-P_1)Y$ given distribution of $𝑌_{nx1}$ ~ 𝑁$(𝜇1, 𝜎^2𝐼)$

I am currently self learning econometrics from A Course in Econometrics of Goldberger and I'm attempting a problem sheet from the course given by my uni. The problem is the following: Let $𝑌_{nx1}$ ~�...
16 views

### Can I trick an ANN into learning individual values of an array from the average?

It is very easy to have an ANN that can compute the average of an input vector, but is it possible to have an ANN that can generate the output vector given just an average of the output vector as the ...
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### Not able to capture spikes in SARIMAX model for future forecast

Data has the spike in 2013 and 2017. To capture this effect in 2021, dummy column is introduced with 1 as spike in a month otherwise 0. Below is the output and the model used, Why the model is not ...
16 views

### what is the best sampling procedure to use? (stratified)

all, i am a little confused on what sampling method to use. i am doing a classification problem where i am trying to classify a person as having cancer or not. there is apparently huge variance in ...
19 views

### Interpretation of the ratio of two pdfs evaluated at a certain point?

What is the interpretation of the ratio of two pdfs evaluated at a certain point? Is that a statistical distance? Is there any applications of it? I only know one application of differential privacy, ...
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### What to choose for null hypothesis and alternate hypothesis? [duplicate]

In this article which explains types of t-tests for “Independent Two-Sample t-test” example: We can confirm that the t-statistic is again less than the t-critical value so we fail to reject the null ...
43 views

### Why can't we accurately compute covariance matrix in high dimensions?

I am reading pg 651 of Elements of Statistical Learning,where is says: "The simplest form of regularization assumes that the features are independent within each class, that is, the within-class ...
189 views

### Mann-Whitney test for a large sample

I'm not a statistician, so pardon me for being naive on this subject. I'm trying to understand if there's any statistically significant difference in the medians of 2 groups. Here are some of the ...
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### Understanding Generalized Canonical Correlation Analysis (GCCA)

I am trying to learn a single embedding from multiple views using Generalized Canonical Correlation Analysis (GCCA) using the below formula. I am confused in the below formula regarding what is to be ...
265 views

### How to combine standard errors for correlated variables

I was wondering what would be the formula for calculating the standard error of a quantity (A) that is the ratio of 2 quantities (A = B/C) if B and C are correlated? According to page 2 of http://www....
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### How to reasonably use z-score time series with goal-setting? [closed]

I have several business groups that periodically report on their individual primary metrics. The metrics differ in units of measure - some are numerically on the order of 10^6, others on 10^2, and so ...
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### Error terms and Residual in Time Series

I started learning Time Series Forecasting and the below questions keep my head confused a lot. Many papers mention that the forecast errors have to be normally distributed and what about the white ...
Let $A, B$ be two uncorrelated random variables with mean $\mu_{A}, \mu_{B}$ and variances $\sigma_{A}^{2}, \sigma_{B}^{2},$ respectively. For some $0 \leq \theta \leq \frac{\pi}{2},$ set \$X_{t}=A \...