Mathematics Stack Exchange
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Questions tagged [statistical-inference]

The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

Statistical inference makes propositions about a population using data sampled from the population. To test a hypothesis about a population, a typical workflow is to select a statistical model of the process that generates the data and then deduce propositions from the model.

Statistical propositions include—

  • a point estimate, which is a particular value that best approximates some parameter of interest,

  • an interval estimate, for example, a confidence interval (or set estimate), which is an interval constructed using a data set drawn from a population so that, under repeated sampling of such data sets, such intervals would contain the true parameter value with the probability at the stated confidence level,

  • a credible interval, which is a set of values containing, for example, 95% of posterior belief,

  • rejection of a hypothesis, or

  • clustering or classification of data points into groups.

3930 questions
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joint probability of correlated normal random variables

Let $Z_1$ and $Z_2$ follow $ N(\delta, \alpha)$ and covariance between them is $\beta$. How to calculate $P(Z_1>0, Z_2>0)?$
Satya Prakash
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Pitman estimator for location paratemer

Given $Y_1,\ldots, Y_n$ are i.i.d random variables with pdf $\ f(y|\lambda) = e^{\lambda - y}$ for $y\geq \lambda$, and $\ f(y|\lambda) = 0$ otherwis, and $\lambda\in (-\infty, \infty)$. Find the Pitman estimator for a location-parameter $\lambda$…
ghjk
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Maximun likelihood estimator

I am trying to calculate the maximun likelihood estimator, but there is something which is wrong and I can't find the error. $f(x;\theta)=\theta x^{-2} e^{-\theta/x}$ for x>0. The likelihood is $\displaystyle\prod_{i=1}^n f(x;\theta) =…
user430110
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95% confidence interval

To determine whether a prep course improved ACT scores, researchers collected a random sample of ACT scores from 1500 U.S. high school juniors who completed a prep course prior to taking the ACT exam. They also collected a random sample of ACT test…
user163862
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How to find the sum of $(x-\bar x)(y- \bar y)$

Please I need help. I need the correct steps how to calculate: Sum $(x-\bar x)(y- \bar y)$? My numbers are: $x$: 2,4,6,8,10 $y$: 3,5,7,10,12 My results are: $\Sigma x=30$ $\Sigma y=37$ $\bar x= 6$ $\bar y= 7.4$ I think $(x-\bar x)^2$ = 40 and…
Sar
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Inferring conclusions from observations

When observing the output of a system simulation, how does one arrive at a reasonable number of observations needed to make an inference about that system? I am observing widgets flowing through a manufacturing system and observing their wait…
gatorback
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How to inference the conditional probability about LDA?

I'm studying the paper of Blei, "Latent Dirichlet Allocation" ( http://www.jmlr.org/papers/volume3/blei03a/blei03a.pdf ). In his paper(page 1003), given equation is $p(\theta, z|w, \alpha, \beta)= \frac{p(\theta, z, w|\alpha, \beta)}{p(w|\alpha,…
KKH
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Determining the degree of freedom for a $\chi$-squared test

I have read that the degree of freedom is calculated by subtracting $1$ from the number of states a random variable can be in. I am performing a goodness of fit test on a $64\times 32$ matrix where the expected frequency of any $a[i,j]$ is $50\,000$…
user328743
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Simple random sample of a Bernoulli and probability function of a statistic.

Let $x_1, \ldots, x_n$ an simple random sample of a Bernoulli distribution of parameter p, $0
Melanctha
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Solving For Joint Likelihood In This Case

Let $Y_1,Y_2,\dotsc,Y_n \sim \operatorname{Poisson}(\lambda)$. Consider $U = \frac{1}{n} \sum_{i=1}^n Y_i$. Given the conjugate Gamma prior distribution $(\alpha,\beta)$, I want to show that the posterior density $\lambda | U$ is a gamma density…
George
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if the variable depends on $\theta$, like $x>\theta$, indicator s.t $I_{(\theta,\infty)}{x}$ be used, what if $x>0$, should I still use indicator?

For finding the minimal sufficient statistic, if the variable depends on the unknown parameter, like $x>\theta$, we should use indicator s.t $I_{(\theta,\infty)}{x}$, but if the variable is just like $x>0$, should I still use indicator? Thanks for…
user133140
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Question about method of moment

According to genetic theory, blood types MM, NM and NN should occur in a very large population with relative frequencies $\theta^{2},\ 2\theta(1-\theta),$ and $(1-\theta)^{2}$, where $\theta$ is the (unknown) gene frequency. (a) Suppose that, in a…
Jakoer
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Confusion regarding the C.I of the mean of some population

I calculated (and verified) the confidence interval for the mean of a population, from this sample: $n=100$, $x_1 = ...=x _6= 36$, $x_7 = ... = x_{17} = 37$, $x_{18}=...=x_{43}= 38$, $x_{44} = ... = x_{75} = 39$, $x_{76} = ... = x_{89} = 40$,…
George
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variance of a sum of independent random variables

I don't get why here https://en.wikipedia.org/wiki/Standard_error, T/n = 1/n²*(n*sig²) Is there a side knowledge to have here ?
Guillaume Paris
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Expectation of an estimator

I want to find the expectation of $\hat\theta$. I have the cumulative distribution of $\hat\theta$: $$\Pr{(\hat\theta>t)} = e^{n(\theta-t)}\quad \text{for $t>\theta.$}$$ Now to find the expectation I need the probability density of $\hat\theta$,…
user2850514
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