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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Transformation of the LR-test for normal data

Given an iid normal sample $X_1,...,X_n$ with unknown mean $\mu$ and unknown variance $\sigma^2$, we want to test $H_0:\sigma^2=\sigma_0^2$ vs. $H_1:\sigma^2\not=\sigma_0^2$ using the likelihood-ratio ...
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19 views

Model selection: differentiating between two biased coins

Suppose you have two "coins" (Bernoulli variables) $X_1 \sim B(p_1)$ and $X_2 \sim B(p_2)$. You're presented with one of the coins at random -- you don't know which, but with probability $p$ it's ...
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31 views

How big should the sample size be to disprove this article?

There is a new poker computer that is claimed to be unbeatable. http://www.theguardian.com/science/2015/jan/08/poker-program-cepheus-unbeatable I beat this computer on my first try today but my ...
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16 views

Creating a ranking system based on multiple variables

I am attempting to create a user ranking system for a trivia game I am developing. Essentially, users are asked a question with only two possible answers to choose from, giving them a 50/50 chance of ...
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1answer
12 views

Sequence of Gamma r.v.s converges in probability to 1

Let $\{X_n\}$ be a squence of Gamma distributed random variables with pdf $$ f(x;\alpha,\beta) = \begin{cases} \hfill \dfrac{x^{\alpha - 1}e^{-x/\beta}}{\beta^{\alpha}\Gamma(\alpha)} ...
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9 views

Mean Squared Error for a Quantized Estimator

Let $P$ be a distribution supported on the unit interval. Suppose we are interested in estimating the mean value (denoted $\theta$) of $P$ based on data. Then if $X_1,\ldots,X_n \in [0,1]$ is data ...
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7 views

From the States Data Set the following output for the Income in 2000 and for the Income in 2010 was created.

From the States Data Set the following output for the Income in 2000 and for the Income in 2010 was created. Which variable has the greatest relative variation? I don't understand what relative ...
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7 views

generating a 2 dimensional normal copuls

Suppose I generate two random vectors with a different configuration (not the same sink) leading to two r.v. $X$ and $Y$. They should be both $\mathcal{N}(0,1)$ distributed. Both vectors are assumed ...
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13 views

deriving frequency distribution from rank ordering distribution of a variable

I have a variable $Y$ whose values can be modelled using the discrete generalized beta function of with rank $r$ of as a parameter. $r$ is rank of a particular value of $Y$. So mathematically it can ...
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31 views

Shortcut for finding variance proof

One of the questions in my textbook asks to prove that $$s^2 = \frac {1}{n-1} \left[\sum _{i=1}^{n}x_i^2 - n\bar x^2\right]$$ where $s^2$ is the sample variance and $\bar x$ is the sample mean. My ...
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1answer
57 views

Precise mathematical definition of Probability.

Is there a precise definition of probability in mathematical terms? For example we have a precise definition of limit in terms of epsilon and delta,so what is the mathematical definition of ...
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17 views

Show that the estimate of a mean is unbiased

Need help with whole question but part (a) first. To show an estimate is unbiased we need E(X) = mu. X = sum of book values + N * mean of D E(X) = N(avg book value) + NE(mean of D) But what is ...
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1answer
16 views

Summation notation with ambiguous subscripts

I'm reading a paper which has the following description; Say we have a time series of correlated sequential observations of the random variable $X$ denoted $\{x_n\}_{n=1}^N$ from a stationary, time ...
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45 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
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3answers
28 views

Is this a conditional probability or not?

Suppose that the telephone calls during one minute time follow a Poisson distribution with mean=4. If people can handle at most 6 calls per minute, what is the probability that the people will receive ...
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2answers
28 views

Bivariate density function

For a distribution function $F$ density function is defined as $\dfrac{\partial^2 F}{\partial x \,\partial y}$. Is it essential that $F$ is differentiable? Is it required that $\dfrac{\partial^2 ...
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17 views

Hypothesis testing method for a single measurement variable but three nominal variables [migrated]

I'm finding it difficult to get a hard and true answer to which test I should use to determine whether the differences in my data are significant. In short: I have 1 measurement variable and 3 ...
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1answer
36 views

Please help me solve average problems

The least and greatest numbers in a list of 7 real numbers are 2 and 20, respectively. The median of the list is 6, and the number 3 occurs most often in the list. Which of the following could be the ...
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1answer
23 views

Algorithm to generate simulated data for any deterministic relation

We have a statistical model (M) that we want to benchmark against simulated data; our argument is that the model M is more expressive than other models in terms of being able to capture all kinds of ...
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1answer
27 views

strong convexity of loss function in multi-dimensional (high-dimensional) space

My question is based on this paper (see the last 10 rows in page 7). It seems this is a general claim: In machine learning or statistic, the loss function $l(W^TX, y)$ (a linear predictor) can never ...
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0answers
16 views

Proofs for PCA, LDA, ICA, HMM learning algorithms and other stuff

I was wondering if there is some kind of encyclopedia of website for all known math proofs. I'm more interested in statistics (PCA, ICA, LDA, Factor analysis, HMM learning, GMM learning) and algebra ...
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27 views

How much data do I need to gather?

I'm studying a manufacturing process that produces widgets, with a number of machines that breakdown or jam occasionally. I want to figure out how many hours of studying I need to do, in order to be ...
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32 views

Fail of reverse implication of completeness. [on hold]

Let $\mathcal{P}'\subset \mathcal{P}$ be two equivalent classes of probability measures on a measure space $(\mathcal{X},\mathcal{B})$, e.g. $\mathcal{P}:=\{P_{\theta} : \theta \in \Theta\}$. Let $T$ ...
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72 views

Expected value of a brownian motion times the deterministic integral of a brownian motion

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, $E (B_t \int_0^tB_s^3ds)$ = ? I try to turn the expected value into a double integral by rewriting the $B_t$ term as 1) $E(\int_0^t ...
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1answer
24 views

What is the meaning ‘uniformly converge’?

Assuming that, we randomly sample $n$ data following a distribution, then if someone claims that the average of these $n$ data uniformly converge to its expectation with rate $O(\sqrt{1/n})$. Here, ...
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34 views

Find the MLE given the following probability

Data $x_1, . . . , x_n$ are modelled as observed values of i.i.d. random variables $X_1, . . . , X_n$ with each having common probability density function given by; ...
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14 views

How to normalize scoring to eliminate bias?

Background: I'm writing an app to allow multiple people to rate an object on a scale of 1-10. There are a few hundred objects to be rated, so I'd like to allow the reviewers to review what they can, ...
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1answer
126 views

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $\sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for an ...
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19 views

Rearrange for solving x (Miller & Siegmund, 1982; equation 8)

I have the following formula from a paper back in 1982 by Miller & Siegmund, "Maximally Selected Chi Square Statistics": α = 0.05 φ() is the standard normal density function: Everything else ...
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16 views

Expected value of kernel density estimator

Wand and Jones in their book "Kernel smoothing" propose estimator for density function $f$ given by term: $$\hat{f}(x; h) = (nh)^{-1}\sum_{i=1}^n K(\frac{x-X_i}{h})$$ Next they compute its expected ...
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19 views

Metric Entropy Upper Bounds

In the paper Information-Theoretic Determination of Minimax Rates of Convergence the authors present Theorem 3 as follows: If $M_2(\epsilon)$ is the $\ell_2$ packing entropy of a density class ...
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45 views

Expected value of an exponential of a gaussian random variable

$$E (Y_t)=E(e^{X_t}) = E(e^{N(X_0e^{at};\frac{b^2}{2a}(e^{2at}-1)}) =\text{ ?}$$ Knowning that $$X_t \sim N\left[X_0e^{at};\frac{b^2}{2a}(e^{2at}-1)\right]$$ $$X_t= aX_t \, dt+b \, dB_t$$ The ...
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8 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let ...
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37 views

non-zero density of a probability density function

I am dealing with some probability density function. Suppose $h_1$ and $h_2$ are standard normal, and $$X=h_1h_2$$ $$Y=h_1^2+h_2^2$$ I am wondering if the conditional random variable $X|Y$ 's ...
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22 views

Interaction term (second order term) always implie the linear term (first order term)? (GLM)

I am doing a GLM model with interaction terms, but the same question can be asked with an ANOVA model. Suppose I have two independent variables $X_1,X_2$ and $Y$ the dependent one. I notice that, ...
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1answer
30 views

Confusion about integration notation

This is probably a silly question but I've never seen this notation: For a > 0, compute $$\int\int_{x/y \leq a} 2e^{-(2x+y)} dx dy$$ What is $x/y \leq a$ there for? This is from my statistics ...
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14 views

survivor functions

The continuous, independent random variables $X_1$,…,$X_N$ have survivor functions $S(x;θ_i )=[{H(x)}]^{θ_i}$ ; $X_i>0$ ; $i=1,...,N$ Where $H(x)$ is an unknown function decreasing from ...
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1answer
41 views

Proof of Central Limit theorem - infinite points

In the following proof I understand that we have taken power n because for sum of variables we take product of characterstic functions. Intuitively I understand why $n \to \infty$ is important but ...
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1answer
81 views

Probability of no students being mechanical engineers

In a class of 100 students, 30 are computer science majors, 49 are mechaincal engineering majors, 13 are civil engineers and the rest are general engineering majors. Assume students only have one ...
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1answer
30 views

Figuring out the distribution of sample variance

If I have a random sample $X_1,...,X_m$ with normal observations where mean $\mu$ and variance $\sigma^2$ then how can I show that $s_x^2=\sum_{i=1}^{m} ...
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1answer
43 views

Why is “random” in the definition of discrete random variable?

We defined discrete random variable as follows: Suppose $S$ is a countable sample space. Then a function $X:S\to R$ is called a discrete random variable. The lecturer made a note that the "random" ...
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19 views

Prediction Intervals for Discrete Data

I would like to generate a prediction interval as described [here]. I suspect that my data comes from a normal distribution, but it was discretized into bins. How would one construct prediction ...
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2answers
62 views

What Percentile Do I fall in with this test score vs average?

I just took an online test and I was notified that the average score for the test was 45% (out of 100%) and I received a 72%. Using these 2 pieces of information would it be possible to calculate ...
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1answer
52 views

Completeness of a statistic. Implication of equivalent probability classes [duplicate]

Let $\mathcal{P}'\subset \mathcal{P}$ be two equivalent classes of probability measures on a measure space $(\mathcal{X},\mathcal{B})$, e.g. $\mathcal{P}:=\{P_{\theta} : \theta \in \Theta\}$. Let $T$ ...
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4answers
53 views

Proofs for statistics and normal distributions

I am studying statistics, and it it, we are given many different results about what kind of estimations we can make and what kind of distributions these estimations have. For example, assume ...
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20 views

Convergence in probability of the Fisher information

Given a family $\{\mathbb{P}_\theta\}_{\theta\in\Theta}$ on $\mathcal{B}(\mathbb{R})$, where $\Theta\subset\mathbb{R}$ and each member of this family is absolutely continuous w.r.t. $\lambda^1$, and ...
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8 views

Integrals involving Marqum Q and Gaussian functions

I want to evaluate the following double integral. Any closed form solution exists? $$I(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-y^2}e^{-z^2} Q_1 \left[\sqrt{y^2+z^2},\sqrt{2x}\right]^2 ...
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18 views

Sampling to find the mode of a discrete probability distribution

I have been trying to find an answer to the following question, that seems annoyingly easy, but every paper I found that somewhat relates to it appears (I am not a statistician by training) to ...
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5 views

How to parameterize some emprical data

I would like to describe a bunch of data that I have collected as a function of two variables. The data is phytoplankton absorption in my local area that has changed in concentration. The data looks ...
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1answer
36 views

Probability distribution of $Y_1=min(X_i)$

Let $Y_1<Y_2<\ldots<Y_n$ denote the order statistics of a random sample of size $n$ from the distribution with pdf : $$f(x;\theta)=e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$ Here we use the ...