Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
5,708
questions
2
votes
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) ...
5
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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{...
6
votes
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 ...
0
votes
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 ...
1
vote
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}]...
0
votes
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, ...
0
votes
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 "...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ā \theta]/\...
1
vote
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 ...
1
vote
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.
...
0
votes
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
$$
\frac{1}{\sigma^{2}} \widehat{S}_{1}=\...
0
votes
1answer
23 views
Need help understanding Bernoulli standard deviation
I'm looking at the solution of a homework problem from a class I took many years ago and I'm not understanding how they calculated standard deviation (see question 1C here: http://datasciencelabs....
2
votes
1answer
33 views
Permutation Hypothesis Testing procedure
I'm not strong in statistics and I'm looking for a help. I work with real estate data and I want to compare apartment prices in 2 districts: district "A" and district "B".
Data ...
0
votes
1answer
24 views
Probability of disease given sensitivity and specificity of test and prevalence [closed]
A disease that occurs in 1% of the population has a test with a 3% false positive rate and a 6% false negative rate. If the test comes back positive for a random member of the population, what is the ...
1
vote
0answers
25 views
How can i test equality of means of two normal populations when $\Sigma$ is known and unknown?
Let's say $\{x_i\}_{i=1}^m,\{y_i\}_{j=1}^n$ are i.i.d samples from two independent multivariate normal populations $N_d(\mu_1,\Sigma)$ and $N_d(\mu_2,\Sigma)$. How can I run a hypothesis test to test $...
0
votes
0answers
10 views
calculate the parameters from a confusion matrix [closed]
am asking how we can calculate the parameters from a confusion matrix:
Confidence Interval =
No information Rate =
P-value =
Mcneamarās P-value=
0
votes
0answers
13 views
Finding partial likelihood for non-proportional model
When the hazard rates for treatment groups are not proportional, then we can use a model with two time-dependent covariates
$$Z_1(t)=\begin{cases}
1 & \text{if chemotherapy only and} ~~t\le ...
12
votes
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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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
votes
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
votes
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:
$$\mathbb{E}_{p(\mathbf{x};\mathbf{\theta})}[f(\mathbf{x};\mathbf{\phi})] = \int p(\mathbf{x};\mathbf{\theta}) f(\mathbf{x};\mathbf{\phi}...
1
vote
1answer
17 views
Find the value that occurs most frequently or use the average?
If I were trying to predict the next number in a list of numbers based on past performance would it make sense to find the number that has occurred most frequently? For example, if I were to say based ...
2
votes
0answers
49 views
How to find variance of a complicated expression?
I have an equation given by
$$
\phi(k)=\sqrt{1-\rho^{2}}\sum_{j=1}^{k-1}\rho^{k-j-1}e(j)
$$
where $\rho$ has value between 0 to 1 and $e$ is modeled as $\mathcal{C}\mathcal{N}(0,\sigma^{2})$, i.e. ...
3
votes
2answers
110 views
Does PCA's reconstruction error get reduced with more PCs being used?
Say that the raw data is $N$-dimensional, where $N$ is a large positive integer. If we apply PCA to the dataset, and compute the reconstruction error (in $\ell^2$-norm) using the first $d \leq N$ ...
0
votes
0answers
16 views
Interpretation of Frobenius norm of covariance matrix?
Is there a statistical interpretation of the Frobenius norm of covariance matrix? More specifically, I have a transformation $Y=T(X)$ where the following holds, what can we say about $T$?
$$\|E[YY']\|...
1
vote
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 ...
0
votes
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
votes
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
votes
0answers
42 views
Residuals may not be normally distributed even if all of the modelās 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 modelās error (noise) terms are.
Is the ...
0
votes
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
votes
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
votes
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:
$$E[f(X)]=\int_A f(...
1
vote
0answers
11 views
how to adjust a result for a variable
Okay. I'm not sure if I'm phrasing it well but here is the problem. I have a set of data which comes out as a density measurement of proteins, these are expressed in a numerical values and correspond ...
3
votes
1answer
65 views
Impossible to overfit when the data generating process is deterministic?
For a stochastic data generating process (DGP)
$$
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
votes
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
vote
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 ...
