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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In what sense is the Jeffreys prior invariant?

I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. Most texts I've read online make some comment to the effect that the Jeffreys prior is "...
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80 views

How to find a symmetric matrix that transforms one ellipsoid to another?

Given two origin-centered ellipsoids $E_0$ and $E_1$ in $\mathbb{R}^n$, I'd like to find an SPD (symmetric positive definite) transformation matrix $M$ that transforms $E_0$ into $E_1$. Let's say $...
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Is Entropy = Information circular or trivial?

I have seen several "maximum entropy distributions" used in the mathematical and statistical literature, often with the justification that they are "minimally informed" beyond the assumptions and data ...
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How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
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390 views

On the probability of obtaining 27 sets in the card game Set

For people not familiar with the card game Set, see its entry on Wikipedia and/or one of the related questions here on Math SE. It might be faster to just play the game a couple of times though, see e....
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Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
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Maximum likelihood and sufficient statistics

$$f_T(t;B,C) = \frac{\exp(-t/C)-\exp(-t/B)}{C-B}$$ where our mean is $C+B$ and $t>0$. so far i have found my log likelihood functions and differentiated them as follows: $$dl/dB = \sum[t\exp(t/C)...
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Unbiased Estimator for a Uniform Variable Support

Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \left[ 0, \theta \right] $. Now I need to estimate $ \theta $ based on $N$ observations and I want the estimator to be ...
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Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
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I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll?

I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll? I am not sure how to set this one up. This one is not homework, by the way, but a question I am making up ...
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How many rolls do I need to determine if my dice are fair?

Roughly how many times do I need to roll a 6-sided die to feel confident that it's giving "fair" results? What about a 10-sided or 20-sided die? Note that I will be actually manually rolling physical ...
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487 views

Probability with unknown variables

An urn contains $10$ red marbles and $10$ black marbles while a second urn contains $25$ red marbles and an unknown number of black marbles. A random marble will be selected from each urn and the ...
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How come in statistics there is very little justification for the formulas used and proofs are almost nonexistent [closed]

I don't understand why people accept certain formulas in statistics without a mathematical proof style argument. You see this a lot in statistics textbooks and unfortunately this spills over with the ...
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Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
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Is a data set really a set?

Originally I thought that in statistics, a data set is just a set of real numbers, and that was it. But in the case of a set, there can only be one instance of any given entry, e.g. in set theory $$\...
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How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula 7.1....
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Probability of getting a job when applying for 3 places

I have the following problem. You apply for jobs and know that if you send your application then every job appointment procedure has two stages: You can be either invited or not invited ...
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A statistical approach to the prisoners problem

Two days ago, I found this problem on reddit (I didn't have access to reddit when I did the math, so I did it with 24 instead of 23, and I decided the warden picked someone every day, not "whenever he ...
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density of sum of two uniform random variables $[0,1]$

I am trying to understand an example from my textbook. Let's say $Z = X + Y$, where $X$ and $Y$ are uniform random variables with range $[0,1]$. Then the PDF is $$f(z) = \begin{cases} z & \text{...
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Proof of the independence of the sample mean and sample variance

I've been trying to establish that the sample mean and the sample variance are independent. One motivation is to try and write the sample variance, $S^{2}$ as a function of $\left\{ X_{2}-\bar{X},X_{3}...
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262 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
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Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
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Proof that the hypergeometric distribution with large $N$ approaches the binomial distribution.

I have this problem on a textbook that doesn't have a solution. It is: Let $$f(x)=\frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}\;,$$ and keep $p=\dfrac{r}{N}$ fixed. Prove that $$\lim_{N \...
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Why does probability need the Axiom of pairwise disjoint events? [duplicate]

I'm a beginning student of Probability and Statistics and I've been reading the book Elementary Probability for Applications by Rick Durret. In this book, he outlines the 4 Axioms of Probability. ...
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Stein's lemma condition

(Apologies if I break some conventions, this is my first time posting!) I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable ...
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Estimate probabilities from its moments

I want to estimate probability $Pr(X \leq a)$, where $X$ is a continuous random variable and $a$ is given, only based on some moments of $X$ (e.g., the first four moments, but without knowing its ...
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What is continuity correction in statistics

Can someone please explain to me the idea behind continuity correction and when is it necessary to add or subtract $\dfrac{1}{2}$ from the desired number (how do we tell whether we need to add or ...
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Great Book on Probability and Statistics (for Computer Scientists)

I'm a Computer Science sophomore and we're studying Probability and Statistics (fundamentals and all). The teacher recommends a book which I don't like since it does not even try and explain ...
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628 views

Monotonic behavior of a function

I have the following problem related to a statistics question: Prove that the function defined for $x\ge 1, y\ge 1$, $$f(x,y)=\frac{\Gamma\left(\frac{x+y}{2}\right)(x/y)^{x/2}}{\Gamma(x/2)\Gamma(y/2)}...
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relative size of most factors of semiprimes, close?

when chatting about RSA a cohort just asserted something like "most prime factors of semiprimes are roughly the same size" measured in bits. ie "bits" is the number of digits in the base2 ...
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What is the intuition behind the generalized confidence interval?

What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution ...
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Average percent increase not equal to total percent increase?

I tried searching around for this but it was difficult to boil down the search terms. Plus nothing seemed to be showing up anyway. What's an easy way to show that the average percentage increase of n ...
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139 views

Heavy-tailed distributions

I have encountered the following two definitions of heavy-tailedness (right tail) for a $[0,\infty)$-valued random variable $X$ satisfying $\mathbb{E}[X]<\infty$: (i) $\limsup_{x\to\infty}\frac{\...
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Lies, damned lies, and statistics

A story currently in the U.S. news is that an organization has (in)conveniently had several specific hard disk drives fail within the same short period of time. The question is what is the likelihood ...
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Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
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1answer
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Expected value and variance of ratio of two sums of two sets of random variables

Let $X_1,X_2,\ldots,X_n$ be iid $\operatorname{Gamma}(\alpha,\beta)$ random variables. Suppose that, conditionally on $X_1,X_2,\ldots,X_n$, the random variables $Y_1,Y_2,\ldots,Y_n$ are independent ...
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If half the population were murderers, and they could only kill once, how many would survive?

So here's the rules: Half the population are murderers Each murderer can only kill once We assume the nobody will fight back, and only murderers can murder Murderers can kill other murderers Only ...
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Can the maximum likelihood estimator be unbiased and fail to achieve Cramer-Rao lower bound?

If some maximum likelihood estimator (MLE) turns out to be unbiased (which does not necessarily holds), then does it achieve the Cramer-Rao lower bound (CRLB) even in finite sample? (It does when the ...
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Arithmetic mean. Why does it work?

I've been using the formula for the arithmetic mean all my life, but I'm not sure why it works. My current intuition is this one: The arithmetic mean is a number that when multiplied by the number ...
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Uniform distribution with probability density function. Find the value of $k$.

For a random sample $X_1,X_2,...X_n$ from a uniform $[0,\Theta]$ distribution, with probability density function $$f(x;\Theta) = \left\{ \begin{array} \ \frac{1}{\Theta} & 0\le x \le\Theta,\\ 0 &...
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Finding $E\left[\frac{\sum_{i=1}^n X_i^2}{(\sum_{i=1}^n X_i)^2}\right]$ of a sample of gamma random variables

Suppose $X_1,\ldots,X_n$ is a random sample from the $\Gamma(k,\lambda)$ distribution where $\lambda$ is unknown and $k$ is a positive integer and known. How can I find $$E\left[\frac{\sum_{i=1}^n X_i^...
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What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6?

Similar to: "What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?" but we allow repeats so 1,1,2,2,3,4,4,4,4,5,5,6 would count. My answer (or simulation) is flawed as I ...
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What does it mean to integrate with respect to the distribution function?

If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the ...
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Coin tosses until I'm out of money

The question I think is a simple one, but I've been unable to answer or find an answer for it yet: There's a simple game: if you flip heads you win a dollar (from the house), but if you flip tails ...
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What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
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Is pi lying on the ground, and on TV?

Consider the leaves from a bunch of trees in a terraced plaza in the Autumn. It may well happen that the tiles of the terrace are squares whose length easily exceeds the length of the stem of the ...
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What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
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Given a data set, how do you do a sinusoidal regression on paper? What are the equations, algorithms?

Most regressions are easy. Trivial once you know how to do it. Most of them involve substitutions which transform the data into a linear regression. But I have yet to figure out how to do a ...
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Prove that the sample median is an unbiased estimator

My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
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Finding expected number of distinct values selected from a set of integers

I have a set of $n$ integers $\{1, . . . , n\}$, and I select three values with replacement. How can I find the expected number of distinct values? Note each value is chosen uniformly and ...