# Tagged Questions

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 analysis.

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### Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
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### Hottest Days of The Year

Recently, there has been much talk in the media of it being the hottest day of the year so far. It has always seemed to me that there are likely many more of these in the northern hemisphere than the ...
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### Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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### Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
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### Exponential is to Poisson as Normal is to ???

In a time series, if the gap between successive events follows an exponential distribution with PDF $\lambda e^{-\lambda}$, then a Poisson distribution with parameter $\lambda$ tells you the ...
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### Probability of number of people who know a rumor

Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
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### Distribution of the difference of two normal random variables.

If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference? I will present my answer here. I am hoping to know if I am right or wrong. ...
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### Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff bound: ...
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### Why is the partition function able to describe the whole system?

No matter what the real system or subject is, if there is a partition function $Z$, then these kind of identities hold $$\langle X\rangle=\frac{\partial}{\partial Y}\left(-\log Z(Y)\right).$$ If one ...
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### Conditional expected value of mutlitple draws from uniform distribution

There are $m$ i.i.d. draws of $x$ made from a uniform distribution on $[0,1]$. The $n$ ($n\leq m$) lowest draws are "winners", i.e. if we write $x_1\leq\ldots\leq x_n\ldots\leq x_m$, the draws $x_1$ ...
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### Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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### Confidence Interval of Information Entropy?

Information entropy, $IE$, is defined as: $$IE = \sum_{i} p_i log\frac{1}{p_i}$$ Where $p_i$ is the probability of event $i$ (and we are summing over all possible events). Let's say I have data ...
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### Who established the word “ Degree of freedom ” in statistics?

I wonder who is the first one that established and applied the word : "degree of freedom" in statistics? Why he/she need degree of freedom in the calculation of many statistical values?