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 find the probability of a specific draw from the urns?
There are 3 urns each containing 10 balls whose colors are either blue or red. In an experiment a ball is drawn from the first urn. If its color is red a second ball is drawn from the second urn. And ...
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P(A or B) given that they are dependent events
Given that an event will occur on the first trial is p and that it will occur in the second trial if it doesn't occur on the first trial is q, what is the probability that the event will occur once? ...
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Tricky probability
The exact problem statement: A car trader usually sells 70% of cars during the last four months of the year and 50% of cars during other months. The car trades are independent and identically ...
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Looking for a bijective transformation of gamma distributed variable such that $ga(\alpha,\beta)$ is transformed to $ga(r,\beta ')$ where $r\in(0,1)$
I have managed to derive the gamma distribution as the sum of $n$ exponentially distributed random variables, ie,
$$ \sum_{i=1}^n X_i \sim Ga(n,\beta=1/\lambda)$$
where each $X_i \sim Exp(\lambda)$. ...
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The limiting distribution of MLE is chi-squared
I have a ML estimator .
How can show that the limited distribution of this ML estimator is chi-squared?
Thanks very much.
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How to solve this Categorical Data Analysis problem? [closed]
In murder trials in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black, ...
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What is the lower and upper quartil of a set of size one?
I have to implement a function to calculate inter quartil range. It should work for arbitrary lists of data. However for edge cases I am unsure how mathematicians actually define the quartil of a set. ...
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sum, max and difference of dependent discrete random variables
I am trying to calculate the following random variable distribution:
$Z= \sum _{i=0} ^{n} [max(X_0, X_1, X_2,...,X_n)-X_i]$
I know $X_i$ probability distribution ($X_i$ are i.i.d. random variables), I ...
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Expectation of $L^2$ norm.
I am reading an article which are estimating the division kernel of a size - structured population.
I have some difficulties in unstanding the line
$$\mathbb{E}[A_3^2]=\left\|\mathbb{E}[\hat{h}_{\ell}]...
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How do I integrate the likelihood of successfully hitting a target as it closes in from a distance with a known jitter (inherent imprecision)?
Very rough C# estimation
https://pastebin.com/tq3d2TEr
It appears that assuming a perfect lock on a non-dodging missile that shootdowns are rather likely. I'll add more to this soon.
Premise
I'm ...
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Definition of probability density
I know that every probability density function (pdf) must be nonnegative and integrate one on $\mathbb R$.
But the definition of pdf requires a distribution function $F$: "we say that a function $...
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Probability of an event occurring at one time of day vs another
I have a dataset of events that occur throughout the year at different times of the day. The events are tallied in the morning and afternoon. I would like to test the probability of the event ...
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How can I determine the needed sample size for populations with a large amount of possible values?
Say there is a set of numbers of size $N$ where $N > 1,000,000,000$, and for each $i$ in $N$: $0\leq i\leq 1000$. In other words, there are over a billion numbers between $1$ and $1,000$.
I need to ...
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Modeling angles and magnitudes using a bi-variate gaussian.
I have a bunch of points in n-d space who's coordinates follow a Normal distribution $(X=x_1,x_2,...,x_n\sim N(0,1) )$.
The coordinates form an angle $\theta$ (with respect to some arbitrary vector $V$...
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What is the best way to make a scale-invariant version of the Wasserstein-1 metric?
I'm calculating the Wasserstein-1 metric to measure the similarity of two univariate distributions. However, it is dependent on the scale of the distributions, as discussed in Is the Wasserstein ...
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bivariate normal distribution mean, standard deviations
Let (X,Y) have the joint bivariate normal distribution with probability density function as
f(x,y)=(1/(15Π))exp{-2[((x-2)²/9))+((y-4)²/25)-((x-2)(y-4)/9]}; -∞<x<∞, -∞<y<∞
What will be ...
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Suppose that $Q$ is an orthogonal matrix and $\bar{Q}$ is the first $j$-columns. Geometrically what is $\bar{Q}\bar{Q}^T$?
The question comes from PCA. What PCA does is that we represent an observation with a new basis $\tilde{Q} = \{q_1, \dots, q_k\}$, then we drop some dimensions in this new basis. In other words, we ...
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Why do variances add when summing independent random variables?
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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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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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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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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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 [closed]
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 [closed]
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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1
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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 [closed]
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? [closed]
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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