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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easy to implement method to fit a power function (regression)
I want to fit to a dataset a power function ($y=Ax^B$). What is the best and easiest method to do this. I need the $A$ and $B$ parameters too.
I'm using in general financial data in my project, which ...
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Negative binomial distribution - sum of two random variables
Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$
How do I go about proving this? I'm not sure where to begin, I'd be glad for ...
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$P(X<Y)$ where X and Y are exponential with means $2$ and $1$
Given that $X$ and $Y$ are independent variables and that $X$ is exponential with a mean of $2$ and $Y$ is exponential with a mean of $1$. Find $P(X<Y)$?
From the information it can be concluded ...
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Minimal sufficient statistics for uniform distribution on $(-\theta, \theta)$
Let $X_1,\dots,X_n$ be a sample from uniform distribution on $(-\theta,\theta)$ with parameter $\theta>0$.
It is easy to show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic for $\theta$ ...
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Maximum of the Variance Function for Given Set of Bounded Numbers
Let $ \boldsymbol{x} $ be a vector of $n$ numbers in the range $ \left[0, c \right] $, where $ c $ is a positive real number.
What's is the maximum of the variance function of this $ n $ numbers?
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How to find unique multisets of n naturals of a given domain and their numbers?
Let's say I have numbers each taken in a set $A$ of $n$ consecutive naturals, I ask myself : how can I found what are all the unique multisets, which could be created with $k$ elements of this set $A$?...
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Order statistics finding the expectation and variance of the maximum
Let $X_1,X_2,\ldots,X_n$ be a collection of independent uniformly distributed random variables on the interval from $0$ to $\theta$.
The question has three parts.
Find the CDF of $F_{x_n}$(x) of $...
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Uniformly Most Powerful Test for a Uniform Sample
Let $X_{1}, \dots, X_{n}$ be a sample from $U(0,\theta), \theta > 0$ (uniform distribution). Show that the test:
$\phi_{1}(x_{1},\dots,x_{n})=\begin{cases} 1 &\mbox{if } \max(x_{1},\dots,x_{n})...
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Boundedness of Radon-Nikodym derivative
I have the following question:
Under which conditions (I mean the conditions on the measures) the Radon-Nikodym derivative is bounded ? I made some researches but I couldn't find any answer, although ...
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What is second Bartlett identity?
I came across the term second Bartlett identity in the wikipedia link: https://en.wikipedia.org/wiki/Variance_function However could not find detail about it. Can anyone help me to understand what ...
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Expected Value of a Hypergeometric Random Variable
How do you show, that the expected value of a hyper-geometric random variable X with parameters $r$,$w$, and $n$ (a box contains $r$ red balls, $w$ white balls and $n$ balls are drawn without ...
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Roll dice and ignore worst results
I roll $n$ dice with $k$ sides each (numbered $1$ thru $k$, laplace).
I then add the numbers of the $m$ best dice (the higher the roll the better).
This sum is the result.
What is the expected value ...
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Consistency of maximum likelihood estimation for Uniform
Let $X_1,...,X_n\sim \text{Uniform}(0,\theta)$. Show that the maximum likelihood estimation (MLE) is consistent.
Setting $Y=\text{max}\{X_1,...,X_n\}$ I know that for any constant $c\in\mathbb{R}$,$$
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How to prove that $\hat\sigma^2$ has $\chi^2_{n-p}$ distribution (linear regression)
Consider the linear regression model:
$$Y_i=r(x_i)+\varepsilon_i\equiv\sum_{j = 1}^p x_{ij} \beta _j + \varepsilon _i,\quad i=1,\ldots,n.$$
where $x_1,\ldots,x_n\in \mathbb{R}^p$ are fixed, $E(\...
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Showing the Clayton Copula is $2-$increasing
This is the definition of a bivariate ($2$-dimensional) copula:
$C(\mathbf{u}):[0,1]^2 \mapsto [0,1]$ is a bivariate copula if
$C(u_{1},0) = 0$ and $C(0,u_{2})=0$; i.e., $C = 0$ if one ...
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