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Mathematics Stack Exchange

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.

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Distributing coins to children

I would like to ask this question, I am quite confused since a long time I haven't used this. A king wants to distribute 10 identical coins among his 4 sons( every distribution is possible, it maybe that one son gets all the coins). He can do this…
Hai Luu
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Probability of receiving a signal

You have a signal receiver that is able to receive two signals, 1 and 2. Both signals have mean 0 and arrive with equal frequency. Signal 1 is normally distributed with variance 4, and signal 2 is normally distributed with variance 9. You receive a…
dphil1
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Is the sample standard deviation asymptotically unbiased as an estimator of $\sigma$?

Let $S_n$ be the sample standard deviation of $(X_i)_{1\le i\le n}$ for a sequence of iid variables $X_i$. It is easily proved (as below) that it underestimates $\sigma$, where $\sigma^2 ={\rm Var} X_i$. In common examples such as the normal…
Jose L. Arregui
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UMVUE of P(x>t) with X~Exp(1/$\theta$)

Let be $(X_{1},X_{2})$ a simple random sample, with $X_{i}\sim\textrm{Exp}(\frac{1}{\theta})$ with $\theta$ unknown. I need to find the UMVUE of $\eta=\mathbb{P}(x>t)=e^{\frac{-t}{\theta}}$, with $t>0$ fixed. I know that $T=\sum_{i}X_{i}$ is a…
Lancet S.
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Probability and Zscores

A company pays it employees an average wage of $ \$15.90$ an hour with standard deviation $\$1.50$ per hour. Assume the wages are approximately normally distributed. What is the hourly wage that is exceeded by only $5\%$ of the employees? Is the…
Lemon
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Is this setup correct? Probability

If observations from an experiment are normally distributed, what percentage of the observations differ from the mean by more than $1.3\sigma$ where $\sigma$ is the standard deviation. Would it be $P(X-\mu > 1.3\sigma)$? Or do I have to look at…
Lemon
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Ask a transformation $Y=2pX$ where X is negative binomial

Let $X$ have the negative binomial distribution with pmf $f(x) = \binom{r+x-1}{x} p^r (1-p)^x, x=0,1,2,...,$ where $00$ is an integer. (a) Calculate the mgf of X. (b) Define a new random variable by $Y=2pX$. Show that, as $p…
Jonathen
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What is the difference between a mosaic chart and a tree-map?

Chart given on test I was given a chart on a test and asked to identify what kind of chart it is. I called it a treemap and I got that answer wrong because the professor said it was a mosaic chart. I was under the impression that those two are…
C1K3
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Comment this proof of Cauchy distribution doesn't have mgf function

In Casella's statistical inference textbook exercise 3.21, it asked the proof to show Cauchy distribution doesn't have mgf. One solution I find online is: The moment generating function would be defined by $\frac{1}{\pi} \int_{-\infty}^\infty…
Jackie
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Prove Rao-Cramer from scratch

I'm trying to prove Rao-Cramer lower bound from scratch. Here's my attemption: It suffice to consider single sample case: $X$ sampled from distribution with p.d.f $p(x;\theta)$. And $f$ is an estimator with $$\int f(x) p(x;\theta) dx=\theta$$ holds…
Peter Wu
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Detect outliers of a single 2D cluster (no normal distribution)

Imagine I have a single cluster of points in a 2D map (not in a normal distribution), each point having (x,y) coordinates. Thus, no need to do data categorization. What's the simplest technique to remove outliers from this 2D cluster? I need to…
João Pimentel Ferreira
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Can likelihood be probability if the models are complete and mutually exclusive

I'm reviewing my thin grasp of likelihood from years ago. For simplicity, let us consider a finite set of candidate models for explaining a set of observed data. They can be completely different models or they can be a parameterized family of…
user2153235
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Methods of Comparing Different Ratios?

I thought of this problem recently. Suppose there is a town with 5 regions. Each of these regions has multiple bakeries - however, people can only visit bakeries within their region. Thus, we know the ratio of customers to bakery in any of these…
stats_noob
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Discrete Data Set, Median, and Calculated 50th Percentile

I'll start by declaring I am no mathematician/statistician whatsoever. So be gentle. I have a data set of 400 observations against discrete values from 1 to 10. Using excel, the median returned a value of 7. I created a histogram with this data set…
David E PMP
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use R to implement likelihood ratio test with given size

Assuming that $X$ has a PDF of the normal distribution form with sd = 1. But we don't know the mean. We want to test the hypothesis $H_0: \theta_0 = 0$ versus $H_1: \theta_0 = \theta_2^*$, and assess the power ofthe test that rejects $H_0$, as…
Jonathen
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