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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Statistics Question

I'm revising for my exams and I want to check if I did this exercise correctly: 10 measurements were done using a certain tool. The average and standard deviations of measurements using a this tool ...
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3k views

What is the difference between empirical distribution , classical probability and axiomatic definition

Can you tell me what is the difference between empirical distribution and classical probability? My teacher has told me that when we take limit empirical distribution will get a constant value ...
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1answer
56 views

What's the probability of a match between a prediction and a measurement?

I have a prediction $f(x)$ of some continuous process variable, based on an input variable $x$ (think: location). The prediction is incorrect, with the error being normal distributed with expected ...
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1answer
279 views

Question on statistics

I have a kind of weird question. But this wont be a harder one. Actually, i feel it is incomplete. I don't have much experience on statistics. But some advance user will be able to understand this ...
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0answers
83 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
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3answers
421 views

Variance of a max function

Say $x_1$ and $x_2$ are normal random variables with known means and standard deviations and $C$ is a constant. If $y = \max(x_1,x_2,C)$, what is $\mathrm{Var}(y)$? Well, I forgot to tell that $x_1$ ...
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1k views

Expected Value of function of two random variable

Assume $X_1$ is an Exponential random variable with unit mean ( i.e. $f_{X_1}(x) = e^{-x}$ ) and $X_2$ is an Erlang distribution with shape $N$ and unit rate ( i.e. $f_{X_2}(x) = ...
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89 views

example of the vector (bounding expectation of a form in Rademacher functions)

Let $x\in R^{2m}$ such that $x_1+\ldots+x_{2m}=0$ and let $r_i, i=1, \ldots, 2m$ be Rademacher functions, i.e. $P(r_i=1)=P(r_i=-1)=1/2$. I would like to find an example of the vector $x$ such that ...
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568 views

Solving this pie-chart

Hi could anyone please guide me on how I would go about calculating the percentage of a specific sector from this Pi-chart. I think I am suppose to use data from the bar chart and apply it to the ...
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2answers
130 views

Identity related to binomial distribution?

While writing a (non-math) paper I came across the following apparent identity: $N \cdot \mathop \sum \limits_{i = 1}^N \frac{1}{i}\left( {\begin{array}{*{20}{c}} {N - 1}\\ {i - 1} \end{array}} ...
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1answer
187 views

Distribution of count data with large spread and heavy concentration of small values

I have a dataset of the counts of each user visiting a set of websites in a year (each user visits at least 1 website in my data). Half of the users visit 7 or fewer sites though the top user visits ...
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11k views

Difference between Probability density function and distribution function?

i am learning for my statistics exam and have to know a lot of theory. My question is: Whats the difference between Probability density function and distribution function? I appreciate your answer!
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2answers
632 views

Cauchy-Schwarz matrix inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ and $Y$ are random vectors, is there a way to bound ...
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1answer
240 views

Gaussian Mixture Model

I am fitting a Gaussian Mixture Model to high-dimensional data (40 dimensions) I have trained the model using EM, learned the parameters and now I want to know quantitatively what is most important in ...
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0answers
300 views

sum with permutations

Let $a$ be vector in $R^{2m}$. And let $S_{2m}$ be group of all permutations on the set $\{1,\dots,2m\}$. I would like to calculate $$ \sup_{\pi\in S_{2m}}\sum_{d(\sigma, ...
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1answer
719 views

Justification for transforming explanatory variables

I am using linear and generalised linear models, and have transformed my explanatory variables using $log10(\bullet)$ and $sqrt(\bullet)$ transformations, and my response variable using an arcsine ...
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0answers
120 views

Windowed Linear Correlation

$\DeclareMathOperator \Cov {Cov}$ $\DeclareMathOperator \Var {Var}$ $\DeclareMathOperator \E {E}$ Consider the following experiment: For $N\geq1$, consider $N$ black balls. Let us paint each black ...
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0answers
53 views

How to quantify the volitility of a time series data?

I am currently working on a time series data and I would like to quantify how volatile it is. Here volatile I mean how "shaky" the series is. If the series is smooth than it is not volatile. I have ...
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1answer
1k views

Sum of truncated normals

Suppose $X_1, \dots, X_n$ are truncated standard normal variables, truncated so that $X_i \geq 0$ (that is, $X_i$ is drawn as a standard normal, conditional on $X_i \geq 0$) Let $c_1, \dots, c_n$ be ...
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2answers
97 views

Ranking students from 2 separate exams in single scale.

Is there a way to rank 2 student groups who face 2 separate exams in a single scale using z-score, given that there are enough student in each group to consider each score distribution a normal ...
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1answer
788 views

What is the difference between FWHH and FWHM?

The title says it all really. I wanted to know if there is a situation where full width half height half maximum is more appropriate than full width half height or vice versa. Thank you
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1answer
318 views

Going from binomial distribution to Poisson distribution

Why does the Poisson distribution $$\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k \exp{(-\lambda})}{k!}$$ contain the exponential function $\exp$, while its relation to the binomial distribution would ...
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1answer
1k views

A problem on random variable in probability

I am a starter in maths. I am doing pretty good in all other topics except for probability. I don't know why I am always confused in it. My exams are nearby and I still cant solve simple problems. Can ...
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1answer
726 views

Median of order statistics

I recently learned that to find the pdf of the median of say $X_1,X_2, X_3$, you first find the Cdf via $$ P(M \le x) =P(\text{at least 2 are}\, \le x) = P( \text{exactly 2 are}\, \le x) + P(\text{all ...
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8k views

Arcsine squareroot transformation for data ranging from -$1$ to $1$

According to the Handbook of Biological Statistics, the arcsine squareroot transformation is used for proportional data, constrained at $-1$ and $1$. However, when I use ...
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2answers
401 views

If the covariance matrix is $\Sigma$, the covariance after projecting in $u$ is $u^T \Sigma u$. Why?

I read in this answer that: If covariance matrix is $\Sigma$, the covariance after projecting in $u$ is $u^T \Sigma u$. I fail to see this, how do I get the covariance of a set of points after ...
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138 views

Random Variables Transformation

A point P is randomly chosen on the triangle with sides' length 1. The triangle is spun randomly (uniformly) about its vertex (0,0). Let (X, Y) denote P's coordinate. Find the joint density of (X, Y). ...
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1k views

Probability density function of a complex-valued random variable

I'm trying to understand the concept of complex-valued random variables, but I'm struggling. If you consider two real-valued random variables $U$ and $V$ with values $u$ and $v$ and the joint random ...
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1answer
350 views

Generating a random monotonically increasing polynomial?

Given a polynomial $y : \mathbb{R} \mapsto \mathbb{R}$ of degree $p$: $$ y(x) = \sum_{k=0}^p c_k\, x^k,$$ can a random set of coefficients $\{c_0, \cdots ,c_p\}$ be generated such that $y$ is ...
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2answers
77 views

Estimation with non-independent errors

I have the following model: $Y_1=\beta+\varepsilon_1+\varepsilon_2$ $Y_2=\beta+\varepsilon_3+\varepsilon_4$ $Y_3=\beta+\varepsilon_1+\varepsilon_4+\varepsilon_5$ ...
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50 views

Estimating Crime

Let's say I have a dataset containing crime statistics for a given city block, how would I go about estimating the probability of a crime occurring to me while I am on that block?
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153 views

Using a Bivariate Gaussian Distribution to Predict Range of Movement

I am currently attempting to use a bivariate normal distribution to identify the most likely range of movement for a blob in computer vision. This itself is not the problem, however; I do not ...
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1answer
66 views

What does the variance/SD of a set signify?

Assuming I have a huge data set and the only attribute I know about it is the Variance (or SD since SD = $\sqrt{\text{Variance}}$). What conclusions can I make about the set with reasonable certainty? ...
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1answer
104 views

Calculate the average number of cards of a certain suit in my opponent's hand

Let's suppose that I am playing a card game with 3 other friends. One of my friends is on my team while the other 2 people are on the opposing team. The cards have just been shuffled and dealt so ...
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124 views

Mathematics of change money

Do you know any results or articles about change money? Something like the statistics of different value notes in a cash box. Or answers to questions which distribution of notes values is best for ...
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358 views

Hyper Birthday Paradox?

There are $N$ buckets. Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets. At $t=1$, we expect that at least one bucket ...
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5answers
1k views

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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4answers
141 views

Please find the expectation

Consider the i.i.d. sequence $X_1, X_2, X_3, X_4, X_5, X_6$ where each $X_i$ is one of the four symbols $\{a, t, g, c\}$. Further suppose that $\mathbb{P}(X_1 = a) = 0.1,\ \mathbb{P}(X_1 = t) = 0.2,\ ...
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50 views

How do I plot a rate?

How do I plot a rate? I've got a rate of exponential growth per year and I want to plot this, and see where it gets to in 10 years, or a thousand years.
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1answer
240 views

Predict the height of a student whose weight is 60 kilograms.

The average height and weight of a group of students turned out to be 5 ft 6 inches and 65 kilograms respectively. The correlation between heights and weights was found to be 0.6. Using the regression ...
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1answer
39 views

Please find the probability

Let X and Y be exponential random variables with parameters 1 and 2 respectively. Another random variable Z is defined as follows. A coin, with probability p of Heads (and probability 1 − p of Tails) ...
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2answers
142 views

Are the square and the maximum of distribution functions a distribution function?

Let $F$ and $G$ be (one dimensional) distribution functions. Decide which of the following are distribution functions. (a) $F^2$, (b) $H$, where $H(t) = \max \{F(t),G(t)\}$. ...
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1answer
94 views

Is it possible to shape Receiver Operating Characteristics?

Given random observations $x$, from a random variable $\mathcal{X}$, we have two different distributions under two hypothesis. \begin{align} \mathcal{H}_0: \mathcal{X}\sim K_1\\ \mathcal{H}_1: ...
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1answer
557 views

Calculating Percentile Rank Using Relative Strength Ranking

I have a spreadsheet of stock quotes that contains Relative Strength Ranking (RSR) ranging from -45 to 65 and the count of ranks is 1500. Could someone please explain how I can calculate percentile ...
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1answer
288 views

question involving Markov chain

Let $S_{2m}$ be the group of all permutations $\pi$ of $\{1, 2, \ldots, 2m\}$. The following transition kernel $S$ generates the random transposition walk $$ Ch(\pi, \pi')= \begin{cases} \frac{1}{2m} ...
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1answer
534 views

Solving a Maximum Likelihood Estimation with an exponential distribution

I need someone's insight on applying a MLE for an exponential distribution. In a finance paper, I have the following: $\displaystyle d_i \sim \frac{\epsilon_i}{\lambda_i}$ where $\epsilon_i$ is ...
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2answers
196 views

Differentiating the posterior distribution function

I am learning about Bayesian statistics and I'm currently doing loss functions. Let $f(\theta | \mathbf{x} ) $ be a posterior pdf . Let $F(\theta | \mathbf{x} ) $ be the associated distribution ...
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203 views

Covariance question: A non squared matrix possible?

I have an academic economic paper that says the following: $$q_r = \operatorname{Cov}(rx,v')\lambda$$ $$(14 \times 1)=(14 \times 4)(4 \times 1)$$ My a vector $q_r$ is of size $14 \times 1$ and my ...
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462 views

Convergence rate of empirical Fisher information matrix

I can see that using the law of large numbers and perhaps mild conditions on the likelihood function, one can show the empirical Fisher information matrix uniformly converges to the true Fisher ...
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1answer
3k views

Mean equal to standard deviation, with positive values

I read the question and the answers regarding the idea of the mean being equal to the standard deviation, in which the mean can always be adjusted to equal anything. But, if the data is restricted to ...