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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.

Statistics is the science of the collection, organization, and interpretation of data. It deals with many aspects of data, which includes the planning of data collection in terms of the design of surveys and experiments. (From Wikipedia)

More specifically, mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and mathematical analysis. (From Wikipedia)

For questions which are more generally about collecting and treating data, it is advised that you post your question on Cross Validated and DSSE.

37109 questions
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Average more accurate than individual guesses

I once saw a documentary where a man enters building with a big jar filled with candies and he would go person to person asking them how much candies they believe were in the jar. The building had little over 200 people. From those 200 none got the…
Murg
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Normal Distribution Or Student Distribution?

If you throw a coin in a vending machine, the coin is being weighed by the machine to determine its value. For statistical purposes, you decide to throw $10$ fifty-cent coins in vending machine A. This results in a sample mean of $7.49$ $g$ and a…
Mark Jacon
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Show $ \sum_1^n (x_i-\bar{x})^2 = \sum_1^{n-1} (x_i -\bar{x}^*)^2+\frac n{n-1}(x_n-\bar{x})^2 $

Let $x_1,...,x_n$ be a random sample from some population with $n\geq3$ and with at most $n-2$ sample points being equal. How can one show $$ \sum_1^n (x_i-\bar{x})^2 = \sum_1^{n-1} (x_i -\bar{x}^*)^2+\frac n{n-1}(x_n-\bar{x})^2, $$ where $…
Fiori_Modena
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Prove that the estimators are biased.

Given the following sentences: Let $X_1,..., X_n$ be a random sample from a $Pois(\mu)$ distribution. Consider the following estimator for $e^{-\mu}=P(X_i=0)$: $T=e^{-\overline{X_n}}$. The independent random variables $X_1;...;X_n$ have a…
TFAE
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Can someone help me with this formula?

I'm writing a software algorithm at the moment which compares survey answers. Questions have $5$ possible answers, and a respondent could choose between 1 and 5 answers. What I'd like to do, for each respondent, for each question, is calculate how…
bodacious
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prove $\lim \limits_{n \to \infty}{\frac{W_n\sqrt{n}}{(n-1)\sqrt{2}}} \sim N_{(0,1)}$

I searched the internet alot . The only relevant clue is in Wikipedia: F-distribution Beside that I didn't find any proof for this theory. If $Y$ has $B\left(\frac{d_1}{2}, \frac{d_2}{2}\right)$ distribution, then show that $X$ with given…
Amin
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Show that $\hat\theta=\frac{2 \bar Y- 1}{1- \bar Y}$ is a consistent estimator for $\theta$

Let $Y_1,Y_2,...,Y_n$ denote a random sample from the probability density function $$f(y| \theta)= \begin{cases} ( \theta +1)y^{ \theta}, & 0 < y<1 , \theta> -1 \\ 0, & \mbox{elsewhere}, \end{cases}$$ Find an Estimator for $\theta$ by using the…
math101
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Is a sample of size $N$ uniquely described by $N$ sample moments?

I have the intuition that a sample of size $n$ should be able to be described uniquely by just N moments. But I don't know if that's true. The idea is that, first, it is obvious that a sample of $n = 1$ can be uniquely describe by just the sample…
Elio Campitelli
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Finding an efficient estimator for $ \beta $ in a sample of $ n $ random variables having the $ \text{Gamma}(\alpha,\beta) $-distribution.

Problem: Suppose that we have i.i.d. random variables $ X_{1},\dots,X_{n} \sim \text{Gamma}(\alpha,\beta) $, where $ \alpha > 0 $ is known. Find an efficient estimator for $ \beta $. Recall that the probability density function of the $…
user45185
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Sufficient statistics for $\lambda$ poisson distribution.

I have the following question out of a book. I even have the solution from solutions manual that I cannot really follow either, so I thought I would ask here to see if someone could dumb it down for me. Let $Y_!, Y_2, ..., Y_n$ denote a random…
Bucephalus
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Sufficient statistic based on a random sample of size n where the pmf is $P_{\theta}(X=x) = c(\theta)2^{-x/\theta}$

so the question asks to find a sufficient statistic based on a sample of size n, where $X \sim f_{\theta}(x)$ and \begin{equation} f_{\theta}(x) = P_{\theta}(X=x) = c(\theta) 2^{-x/\theta}, \quad x = \theta, \theta+1,..., \theta…
Andrew Liu
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Question of maximum likelihood estimation.

A population density function is definded as $$ f(s)=\begin{cases} (W-1)s^{-W} & s\geq 1, \\ 0& \text{elsewhere}, \end{cases} $$ where $W>1$ is unknown. I just want to ask, how do i find the maximum likelihood estimator of $W$? Do I do the normal…
dorothy
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Why is there a linear relationship between the quantiles of a generic normal distribution and those of the standard normal distribution?

I don't understand why a quantile-quantile plot is linear. That is, if you plot quantiles of the normal distribution with mean $\mu$ and standard deviation $\sigma$ (on the vertical axis) against quantiles of the standard normal distribution (on the…
user7661
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Proving if $X$ is a random variable and $f$ is a continuous function, then $f(X)$ is a random variable

I'm reading "Fundamental of Mathematical Statistics" by Gupta and Kapoor and the authors make the claim that "if $X$ is a random variable and $f$ is a continuous function, then $f(X)$ is a random variable". The proof has been skipped as it was…
ashK
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Basic Statistics Question (sample, normal distribution)

I am working on a question for an econometrics class that involves using the program Stata. It is as follows Suppose $X_i$, $i=1,2,...,n$ are i.i.d random variables, each distributed $N$($19,9$). Define $\bar{X}$ to be the mean value of the $n$…
clarence_atwater
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