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.
35,573
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Why do variances add (when summing distributions)?
I understand intuitively why spread would be additive, but not precisely why variances add rather than, say, the standard deviations.
And how does this relate to the pythagorean theorem/euclidean ...
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Quantitative analysis Statistical test of Likert Scale to determine best option from multiple options
What is the best statistical test to be done to determine the best drawing of three drawings?
Given data collected from a Likert scale on various characteristics of the drawings. Questions on ...
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How to determine number of entries in buckets of a perfect normal distribution?
Say you have a sample (n = 50,000) of student test percentages that range from 0% to 100%. If I create 5 buckets [0,20) [20, 40)..., how would I figure out the number of students to put in each bucket ...
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How is $q_{\alpha,k,N-k}$ in Tukey's test calculated?
On the wikipedia page for Tukey's range test, under section Confidence Limits, a quantity
$$q_{\alpha,k,N-k}$$
makes an appearance. Unfortunately, it is not defined ...
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If sample average converges almost surely in an iid sample, must it converge to the mean?
SLLN tells us that if $X_1,...,X_n$ are iid, with $X_1$ having finite mean $\mu$, then their sample average converges a.s. to $\mu$.
However, suppose instead we know that $X_1,...,X_n$ are iid and ...
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The problem of probability density tails by KNN
There is the sample with 15 values (ranged): {315, 319, 319, 320, 325, 326, 326, 327, 327, 328, 328, 330, 331, 332, 336}.
The KNN method is used for probability density estimation. Let the estimator ...
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Probability and Random Variables.
Hi, I was trying to understand this example in the book. In the first part of the question, We've to find p.d.f (probability density function). For that, we take the derivative of the given ...
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Deriving Properties of Estimators (Bias and Variance)
I have the following probability distribution function given by:
\begin{equation}
\label{eq:function}
f(x) = \frac{4a}{x^5} \exp \left[ {- \frac{a}{x^4}} \right] \quad \quad 0 \leq x \leq \infty \...
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2
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Variance of min of r.v. and constant
I am not a student of statistics, but need to compute an expression for my work. This is what I have so far:
I have a r.v. $D$ (pdf: $f$, support: $[0,\infty]$), and a positive constant $q$. I have a ...
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Order Statistic as a Consistent Estimator
Problem. Let $X_{1},\ldots,X_{n}$ denote a random sample from the distribution with common pdf
$$ f(x;\theta) = e^{-(x-\theta)} 1_{(\theta,+\infty)}(x), \;\; \theta \in \mathbb{R} $$
Let $Y_{n} = \min ...
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The difference between the sum of the squares of the diagonal elements of WAW (for Wishart matrix W) and matrix A （for any A）
For any symmetric matrix ${ A}\in R^{K\times K}$, we can compute ${ B} = { W}{ A}{ W}$ where $W\in R^{K\times K}$ is a Wishart matrix with $N$ degrees of freedom (N>K).
I want to bound term $\sum_{...
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Is there any references for solving inverse Ising problem w.r.t. some objective functions other than MaxLikelihood
I am trying to formulate an inverse Ising problem that optimizes some defined objective functions other than maximum likelihood. I am pretty new to this field (only some background on Markov random ...
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Deriving the log likelihood of the observed data
Hi, I am deriving the log likelihood of the observed data in part a.
How do I derive it?
Below is my solution:
$$Y_i \sim N(\mu, \sigma^2) = f(Y_i = x_i) = \phi (\xi; \mu, \sigma^2)$$ (for $r_i =1$)
$$...
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Prove that there does not exists an estimator for which the risk is $0$.
Let $\lbrace \mathbb{P}_\theta \rbrace_{\theta\in \Theta}, \Theta \subset \mathbb{R}$, be an identifiable parametric family of distributions with common support, where card$(\Theta)\geq 2$. Consider ...
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The estimator of the capability indices
The capability indices $C_{pk}$ and $P_{pk}$ are defined for a normally distributed random variable $X$ with mean $\mu$ and standard deviation $\sigma$ and specification limits $-\infty <LSL < ...
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Show Gateaux differentiability
The partially linear model, $\theta_0$ and $g_0,m_0 \in F\subseteq \{f|f:\mathbb R^d \rightarrow \mathbb R\}$ where $F$ is a fixed function class, and we consider a random vector $Z=(D,X,Y) \in \...
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Can $(1-10^{-x})^y$, where x and y > 0, be simplified?
I would like to know if it is possible to simplify the expression $(1-10^{-x})^y$ where $x$ and $y\gt0$. If yes, the simplified formula would be helpful.
I don't have the competence or the tools (e.g. ...
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The complete sufficient statistics for uniform distribution
Given uniform distribution $(0, \theta)$, we know the complete sufficient statistics is $X_{(n)}$. We can show it is complete by definition of the completeness.
Another uniform distribution is $(\...
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Bound on the expected time of first success in a series of Bernoulli RVs
Given an infinite series of Bernoulli RVs $X_1,X_2,...$ (which may be differently distributed and mutually dependent), we are given that for every $n>0$, $$\mathbb{E}\left[\sum_{t=1}^{n}(1-X_t)\...
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Proof to show that zero covariance implies independence for two Bernouilli-distributed variables
I want to prove that if $X,Y$ are Bernoulli-distributed (with $p_1, p_2$ for parameters), and if $Cov(X,Y) = 0$ then $X$ and $Y$ are independent. My proof is the following :
$Cov(X,Y)=0 \iff E(XY) = E(...
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MLE for the sum of a normally distributed variable and constant after a specific time [closed]
We start off with a normally distributed random variable $X$ with known $\mu=100$ and $\sigma^2=1$, and after $\vartheta$ days, a constant $1$ gets added to the value each day. Given $X_1,...,X_n$ ...
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Is the pursuit of asymptotic variance for a novel estimator always required even if the estimator itself is complicated?
I am working on an academic paper proposing a new estimator for a population mean. It works quite well in simulations across various superpopulation models (linear, quadratic, and strictly nonlinear). ...
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Statistics random variable change
I have a question about my statistic homework. We have to generate n=100,500,1000 random numbers uniformly distributed on the (0,1) interval with rand() function . And i have to generate n values of ...
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Positive and negative skewed distributions
I have seen the following statements in several places:
In a normal distribution, mode=mean=median.
In a negative/left-skewed distribution, mode > median > mean.
In a positive/right-skewed ...
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The two random variables $X$ and $Y$ have the following common distribution
EDIT: here is the formatting of the problem. I already solved a) and b), I attempted to solve c) and struggle with d)
$$
\begin{array}{c|lcr}
X \setminus Y & 0 & 1 \\
\hline
-1 & 0 & ...
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How to show $\mathbb E[f(X)]=f(\mu +i\sigma)$ using Cauchy integral formula
Assume that $C(\mu ,\sigma )$ is the Cauchy distribution with location $\mu \in \mathbb{R}$ and scale $\sigma >0$ and the density function of $C(\mu ,\sigma)$ is $p(x;(\mu ,\sigma))=\frac{\sigma}{\...
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What is the probability that there will be 1 ministerial position with two claims, 1 position with no claims, and 8 positions with one claim?
I have a question regarding a counting problem:
a)Within the coalition of five parties ten ministerial positions must be divided between the parties. Each party is allowed to claim two such positions, ...
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Given this weak property is it possible to demonstrate that the difference of expected value is negative?
Let's assume that we have $X,Y$ as random variables and we have as hypothesis that
$$X-\mathbb{E}_x \leq Y-\mathbb{E}_y$$
where $\mathbb{E}_x$ is the expected value of x.
Is it possible to demonstrate ...
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Solving a simple equation of Allelic richness (diversity) [Population genetics] [closed]
Dear Math SE Community,
I would like to manually calculate the following equation on my simplified data. First, this is my data:
I have two sample (A and B) and for this two samples I have 3 loci. ...
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Posterior for Pareto type II (Lomax) likelihood with a Jeffrey's prior on tail index
For Pareto Type II (Lomax) $X_i\sim\text{Lomax}(A,\alpha)$, the Jeffrey's prior is still
$\frac{\sqrt{n}}{\alpha}$
The joint is
$
\frac{
\sqrt{n} A^{n\alpha}\alpha^{n-1}
}
{
\prod _{i=1}^n {{\left(A+...
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Gaussian Bell Curve
Is the Gaussian Bell Curve time dependent?
Suppose we toss a coin and depending on the outcome we win or lose one dollar. If we do it for a long time (infinity), the outcome when plotted will resemble ...
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Statistics question about probability
Im trying to study for my statistics exam and Im having trouble with this question.
"Imagine you wished for 2 christmas gifts.The chance of getting J1 is 0,8. The chance of getting J2 is 0,2 if ...
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Normal approximation of the ratio of Normally distributed random variables sums
Given $n$ independent Normally distributed random variables $X_i \sim N(\mu_i, \sigma^2_i)$ and $n$ real constants $a_i \in \mathbb{R}$, I need to find an acceptable Normal approximation of the ...
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Find the distribution of Q
Let $X_1, X_2,..., X_9$ denote a sample. Assume that $X_i \sim N(4\theta, \theta^2)$ for $i = 1,...,9$ with an unknown $\theta > 0$. We want to find the confidence interval for $\theta$.
a) Find ...
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Can somebody offer suggestions rooted in mathematical theory as to how many data points I require for a regression line to accurately model an object?
Essentially for a math paper I have to apply regression to model the outline of an object. In order to do so I go on GeoGebra and use the "plot point" function to plot data points along the ...
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What does a percentile of 8th–22nd mean? [closed]
openai unveiled some details of gpt-4, here's a piece of it.
Does a percentile of 8th–22nd here mean 14% of candidates scored 2 points?
Assume there were 100 students in total, including gpt-4, took ...
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$\mathbb E[X^2]$ vs $\mathbb E[X]^2$ in Statistics
In a statistics problem solution it stated:
$$ \mathbb E[\bar{X}]^2 = \operatorname{Var}(\bar{X})+ (\mathbb E[\bar{X}])^2 $$
I remember that
$$ \operatorname{Var}(X) = \mathbb E[X^2] - (\mathbb E[X])^...
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Confidence Interval and Hypothesis Test Disagree?
I created a problem where I am having my students complete a hypothesis test and confidence interval for proportions to highlight the connection between the two. However, in doing the problem I am ...
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Description of levels of measurements mathematically
In statistics, i have Omega as the probabily set or X is the random variable. I have now heard there are levels of measurements like here:
How is a nominal scale defined mathematically? Is there a ...
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How to calculate the average of two numbers from two percents [closed]
I am trying to figure out the average IQ of a population if 6% has an IQ of 97 and 94% has an IQ of 100.
I figured that you could maybe do 16.666 x 100 + 97 then divide by 17.666 and get 99.83
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On the autocovariance function of progressively measurable processes
Suppose I have a progressively measurable process $G_\omega(t)$ (adapted process + càdlàg paths), which satisfies
$$\tag{1}
\mathbb{E}_\omega\left(\int_{t_0}^t(G_{\omega}(t'))^2\mathrm{d}t'\right)<+...
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Determining type of statistical test to be used to compare mean hours of sleep for two groups
I am conducting a study on the sleeping habits of students in social sciences vs science. Specifically, I want to compare the means of hours of sleep between the two groups, both for the whole week ...
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KL divergence for distribution representing sums of iid random variables
Sorry if my description is inaccurate, I hope it's understandable.
Given $X_1,...,X_n$, a series of $n$ iid Bernoulli RVs with means $p$, and a similar series $Y_1,...,Y_n$ with means $q$, we know ...
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Find bounds on $\mathbb{E}[X]$
Question: Suppose $X$ is a random variable with $\text{img}(X) = [0,10]$. If $\mathcal{P}(X > 5) \leq 0.4$ and $\mathcal{P}(X < 1) \leq 0.5$, what are the minimum and maximum possible values of $...
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Using z-scores to estimate student performance - valid or not?
I want to know if using $z$-scores is a valid way to predict/estimate a student's exam result given that the student was unable to complete the exam.
Say the student sits exam A:
Result: $88\%$
Exam ...
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What is meant by $\textbf{P}(X < x)$ in statistics? [closed]
What is meant by $\textbf{P}(X < x)$ in statistics?
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Estimating $\lambda$ in a Poisson Distribution from a set of data
I need to estimate $\lambda$ from this data.
The observed frequencies / probabilities are obtained by doing each total number observed divided by $280$.
I know that $P(X=0) = \frac{e^{-\lambda}\cdot \...
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Is the chi squared goodness of fit test valid for the multinomial distribution if the outcomes are unlabeled?
I'm investigating Sketchy Dice Inc. for purportedly selling unfair dice with the following probability distribution:
\begin{align}
p(1)&=0.1 \\
p(2)&=0.3 \\
p(3)&= 0.2 \\
p(4)&= 0.15 \\...
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Calculation of Autocorrelation Function in a System with Harmonic Oscillations in a Fluid: A Study of Langevin Dynamics
Introduction
Explanation of the system being studied
Objective of the study
Presentation of the Langevin equation governing the dynamics of the particle in the system
I am studying a system that ...
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0
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Variation of fibonacci? [closed]
Somewhere I stumbled upon a following method/function. I initially thought it is a recursive fibonacci function, but it is not. Any idea what this function do or what would be its purpose?
...
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