Questions tagged [mathematical-statistics]

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

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2
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3answers
65 views

Getting the variance of $X$ from $Var(\ln(X))$

I've a multiplicative model for which $Y = X*Z$, for which $Y$ and $Z$ is known. I want this model to be additive (therefore using logarithms) to figure out the variance of $X$. I have $$ln(Y) = ln(X) ...
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1answer
42 views

How close to being memoryless can you make a distribution with bounded support?

Related to Exponential-like distribution with support [0,1] I wondered just how close to memorylessness a continuous distribution with bounded support can get. For a continuous variable to be ...
0
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1answer
8 views

Running Logit Model on data and changing weights

I have created a logit model off of my data. There are two independent variables and dependent is 0/1. If I want to run my model off of a dataset I can create probabilities from the coefficients and ...
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1answer
13 views

Interaction and nested structure in symbolic formulae

I am using the following two structures, one simpler and one more complex, in lme4 format which, as you can see, have an interaction term in the fixed effects as ...
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0answers
10 views

Estimating a network-like structure with lead/lag relations

This is a very general question, but any ideas or hints towards the right direction would be very helpful. Assume a dataset with a number of countries $i = 1, \ldots, N$ and a set of indicators $X \in ...
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1answer
28 views

same cdf equals same expectation?

So, if $X$ and $Y$ are both continuous random variables with the same cdf, does that mean that their expectations are the same? And the same thing in case $X$ and $Y$ are both discrete. Thanks in ...
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3answers
34 views

What are the differences between the likelihood functions in Maximum Likelihood Estimation and in the Bayes' Theorem? [duplicate]

I am wondering the differences between the likelihood function in Maximum Likelihood Estimation and the likelihood function in Bayes' Theorem. To me, the likelihood function in Bayes' Theorem depends ...
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0answers
36 views

Probability of a unit being selected in sample

I recently studied sampling from one of online courses and got to know about various sampling methods but i'm confused regarding the probability of an element being selected in a sample. In that ...
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1answer
18 views

How to prove a variable has a log-normal distribution knowing that the variable is a function of a normal random variable?

Let $X$ be a normal random variable with mean $\mu$ and variance $\sigma^2,\; X\sim N(\mu, \sigma^2).$ Prove that the variable $Y = \exp(X)$ has a log-normal distribution. $$f(y)=\frac{1}{y\sigma\sqrt{...
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2answers
314 views

Probability that linear combination of normal random variables exceeds a value

I'm new to statistics, and I'm wondering how can I compute $P(2Y_1 + 4Y_2 - 3Y_3 \geq 40)$ given the following information? $Y$ follows a multivariate normal distribution. $Y_1, Y_2, Y_3$ follow a ...
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0answers
8 views

How to estimate when some values have already stabilized?

I am benchmarking some AI models. The first inference takes more time because it also measures the time to send data to the TPU's memory. Something like this: TIME TO SEND THE TENSORS TO THE TPU'S RAM ...
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0answers
22 views

why is the bias of an AR(1) model converging towards 0 for $n \rightarrow \infty$

could someone please explain to me why the statement at the end is true? The estimator of $\alpha$ in an AR(1) process is biased, meaning: $E[\hat{\alpha} ]\neq \alpha$ this is because $E[\hat{\alpha}]...
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0answers
17 views

Calculating a processes efficiency that has inconsistent yield

This related to programming, but the question at hand is purely a mathematical/statistical one I being ask to display an efficiency of a process over a set period of time. Normally not a an issue, ...
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1answer
19 views

Shape of derivative with respect to vector

I'm confused about the shape of the derivative with respect to vector. In the book MATHEMATICS FOR MACHINE LEARNING page 150: For formula 5.56a, x1 to xn are horizontal. For another pdf "...
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0answers
35 views

Hypothesis testing for mean difference of 2 samples

For example, \begin{array} {|r|r|}\hline & \text{Sample}\enspace X & \text{Sample}\enspace Y \\ \hline mean & 14 & 20 \\ \hline median & 5 & 5 \\ \hline \end{array} How ...
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0answers
28 views

Understanding auction types

I'm working on a project in the advertising space where advertisers bid on inventory. I'm trying to simply understand a practical example of the different auction algorithms. I have spent several days ...
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0answers
32 views

Extreme Value (Gumbel) distribution a member of exponential family

This is a question for discussion in my Linear Model class. I am having a hard time showing that the distribution belongs to the exponential family PDF: $f(y; \theta) = 1/\varphi \exp([y − \theta]/\...
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0answers
17 views

How to prove that the Ys are independent under the normality assumption of $\epsilon$

I am working on a self-study of Cosma Shalizi's "The Truth About Linear Regression". I am currently work my way through Chapter 5, "The Method of Maximum Likelihood for Simple Linear ...
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1answer
10 views

Statistical test - pre and post teaching session

I am collecting data on people's knowledge of radiation dosages before and after a teaching session. The data is NOT normally distributed. Therefore I will be comparing the MEDIAN of each set of data. ...
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1answer
20 views

Show change of expression of sample variance and explain the distribution

Show that $$ \sum\left(Y_{i}-\mu\right)^{2} / \sigma^{2}=(n-1) S^{2} / \sigma^{2}+\left[(\bar{Y}-\mu)^{2} n / \sigma^{2}\right] $$ can be changed into a form $$ \frac{1}{\sigma^{2}} \widehat{S}_{1}=\...
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1answer
23 views

Need help understanding Bernoulli standard deviation

I'm looking at the solution of a homework problem from a class I took many years ago and I'm not understanding how they calculated standard deviation (see question 1C here: http://datasciencelabs....
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1answer
33 views

Permutation Hypothesis Testing procedure

I'm not strong in statistics and I'm looking for a help. I work with real estate data and I want to compare apartment prices in 2 districts: district "A" and district "B". Data ...
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1answer
24 views

Probability of disease given sensitivity and specificity of test and prevalence [closed]

A disease that occurs in 1% of the population has a test with a 3% false positive rate and a 6% false negative rate. If the test comes back positive for a random member of the population, what is the ...
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0answers
25 views

How can i test equality of means of two normal populations when $\Sigma$ is known and unknown?

Let's say $\{x_i\}_{i=1}^m,\{y_i\}_{j=1}^n$ are i.i.d samples from two independent multivariate normal populations $N_d(\mu_1,\Sigma)$ and $N_d(\mu_2,\Sigma)$. How can I run a hypothesis test to test $...
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0answers
10 views

calculate the parameters from a confusion matrix [closed]

am asking how we can calculate the parameters from a confusion matrix: Confidence Interval = No information Rate = P-value = Mcneamar’s P-value=
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0answers
13 views

Finding partial likelihood for non-proportional model

When the hazard rates for treatment groups are not proportional, then we can use a model with two time-dependent covariates $$Z_1(t)=\begin{cases} 1 & \text{if chemotherapy only and} ~~t\le ...
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5answers
2k views

What if a non-random sample is identical to a random sample?

Sometimes, in political polls, pollsters take non-random samples from a given population, but then they apply the results of the theory of random sampling to their non-random sample. I've heard ...
0
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1answer
28 views

Checking the normality and assumptions of residuals in a regression model with a categorical IV

I have conducted a hierarchical regression with 2 categorical variables. One of which I am controlling for (ethnicity-dummy coded). I need to check the assumptions of normality, linearity and ...
0
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2answers
68 views

How to calculate joint pdf of two normals?

Say $(X_1,X_2,X_3)^T \sim N_3\left(\pmatrix{3\\1\\4}, \pmatrix{6&1&-2\\1&13&4\\-2&4&4} \right)$. What is the joint pdf of $Y_1$ and $Y_2$ if $Y_1 = 2+X_1+X_2+X_3$, $Y_2 = 5+...
0
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1answer
57 views

Motivation for Confidence Intervals

I am using the following example to motivate the use of confidence intervals and to better understand them. However, I am struggling with some basic definitions and would appreciate some guidance. I ...
0
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0answers
30 views

Determine whether an error is random?

We have two measurements of the temperature of the same object as: $T_1 = 46 \pm 2 F$ and $T_2 = 58 \pm 6 F$. Normally, the agreement between the measurements could shown to a standard deviation by ...
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0answers
37 views

How to mathematically describe convergence of sample density to population density?

Suppose we have a normally distributed random variable $X$ representing some population. Suppose we draw $n$ samples from the population: $$ (X_1,X_2,\dots,X_n). $$ This can be done using the ...
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0answers
31 views

What is the name of the probability distribution of ROC-AUC when training machine learning models?

When training ML models like neural networks they are random initialized. That has the effect that the results (ROC-AUC for example) are influenced by random effect. When I train them multiple times ...
0
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3answers
36 views

Why does pre-summing variables lead to a different OLS fit?

Suppose I have an OLS model like this: $$y = \beta_1x_1 + \beta_2x_2 + \epsilon$$ If you sum the variables first, $x_3 = x_1 + x_2$ and fit $$y = \beta_3x_3 +\epsilon$$ I expect that $\beta_3$ is the ...
4
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0answers
50 views

What does it mean to take the expectation with respect to a probability distribution?

I see this expectation in a lot of machine learning literature: $$\mathbb{E}_{p(\mathbf{x};\mathbf{\theta})}[f(\mathbf{x};\mathbf{\phi})] = \int p(\mathbf{x};\mathbf{\theta}) f(\mathbf{x};\mathbf{\phi}...
1
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1answer
17 views

Find the value that occurs most frequently or use the average?

If I were trying to predict the next number in a list of numbers based on past performance would it make sense to find the number that has occurred most frequently? For example, if I were to say based ...
2
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0answers
49 views

How to find variance of a complicated expression?

I have an equation given by $$ \phi(k)=\sqrt{1-\rho^{2}}\sum_{j=1}^{k-1}\rho^{k-j-1}e(j) $$ where $\rho$ has value between 0 to 1 and $e$ is modeled as $\mathcal{C}\mathcal{N}(0,\sigma^{2})$, i.e. ...
3
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2answers
110 views

Does PCA's reconstruction error get reduced with more PCs being used?

Say that the raw data is $N$-dimensional, where $N$ is a large positive integer. If we apply PCA to the dataset, and compute the reconstruction error (in $\ell^2$-norm) using the first $d \leq N$ ...
0
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0answers
16 views

Interpretation of Frobenius norm of covariance matrix?

Is there a statistical interpretation of the Frobenius norm of covariance matrix? More specifically, I have a transformation $Y=T(X)$ where the following holds, what can we say about $T$? $$\|E[YY']\|...
1
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1answer
51 views

Testing $A\mu + b = 0$ for a normal distribution

Say $\{x_i\}_{i=1}^m$ are i.i.d observations from a multivariate normal distribution $N(\mu, \Sigma$). How could I test the hypothesis $H_0:A \mu + b=0$ against $H_0:A \mu + b\neq0$, such that A is ...
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0answers
11 views

Survival analysis with continuous outcomes

I'm thinking of running survival analysis for continuous binomial outcome. All instances start at 0 and end at n. For example: ...
0
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0answers
6 views

how to sample a best ad from multiple ad click distuibutions? [closed]

There are 3 ads A, B, C. The probability that users will click the ad is following independent Bernoulli distribution. The history of click are as following ...
0
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0answers
42 views

Residuals may not be normally distributed even if all of the model’s error (noise) terms are?

Hi I have a TURE or FALSE question below: In simple linear or multiple regression, each of the residuals may not be normally distributed even if all of the model’s error (noise) terms are. Is the ...
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0answers
21 views

Problem with some Normal Distribution related questions

so I'm familiar with normal distributions but not when one random variable W is conditional on Z and both are normally distributed. I've attached a picture of the questions. I can imagine the concepts ...
4
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3answers
116 views

Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods

I was recently reading a discussion amongst mathematicians/statisticians about machine and deep learning, and how they are applied by non-mathematicians/statisticians. The argument was that these ...
7
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2answers
296 views

Understanding different Monte Carlo approximation notations

Currently working on a project involving Monte Carlo integrals. I haven't had any prior studies of this method, so hence the following question. Consider the following expectation: $$E[f(X)]=\int_A f(...
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0answers
11 views

how to adjust a result for a variable

Okay. I'm not sure if I'm phrasing it well but here is the problem. I have a set of data which comes out as a density measurement of proteins, these are expressed in a numerical values and correspond ...
3
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1answer
65 views

Impossible to overfit when the data generating process is deterministic?

For a stochastic data generating process (DGP) $$ Y=f(X)+\varepsilon $$ and a model producing a point prediction $$ \hat{Y}=\hat{f}(X), $$ the bias-variance decomposition is \begin{align} \text{Err}(...
21
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5answers
4k views

Why don't linear regression assumptions matter in machine learning?

When I learned linear regression in my statistics class, we are asked to check for a few assumptions which need to be true for linear regression to make sense. I won't delve deep into those ...
1
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1answer
42 views

Book recommendation reqd - A book that teaches both, fundamentals of statistics and the related mathematical concepts, in a comprehensively [duplicate]

I am trying to build a foundation for data science through self study. Is there a book that could teach me the fundamentals of statistics along with the related mathematical concepts in a ...

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