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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Finding a surface fitting equation for this set of data

I have a question regarding surface fitting (3D curve fitting), and since I am not from a maths/stats background I was wondering if someone can help me or point me to the right resources? I had ...
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1answer
280 views

Log likehood functions - Expected value

Let $X_1,X_2,\ldots,X_n$ be a random sample from a Bernoulli($θ$) distribution with probility function $$P(X=x)= (θ^x)(1-θ)^{1-x},\qquad x=0,1;\ 0 < θ < 1.$$ $dl/dθ = [n \overline{x}/θ] \cdot ...
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156 views

Creating somewhat-normal distributions using random numbers - Why does this work?

I'm a software engineer. I asked a friend for help easily creating a somewhat-normal distribution. This is just for the purpose of adding entropy to an application. The code he gave me is this: ...
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86 views

Multiple regression with model $Y = (1 + c_1X_1)(1 + c_2X_2)\ldots(1 + c_nX_n)$

I'm currently working with data contained in $Y, X_1, X_2, \ldots, X_n$ and wish to fit it to the model: $Y = (1 + c_1X_1)(1 + c_2X_2)\ldots(1 + c_nX_n)$ where the $c_i$ are coefficients to be ...
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35 views

Weighting dataset by one of its fields

I'm looking at the Olympics medal count rankings, I've created a points system for medals (Gold - 3, Silver - 2, Bronze - 1). Could someone please tell me how I can weigh the resulting points by ...
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1answer
172 views

Unconfounded assumption

In the notation of the unconfounded assumption, does $$\left(Y(0),Y(1)\right)\perp W \mid X $$ mean $$ f(Y(0),Y(1), W\mid X)=f(Y(0),Y(1)\mid X)\cdot f(W\mid X)$$ ? I can prove that the second line ...
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1answer
57 views

What is the chance of an event happening a set number of times or more after a number of trials?

Assuming every trial is independent from all the others and the probability of a successful run is the same every trial, how can you determine the chance of a successful trial a set number of times or ...
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2answers
151 views

Hypothesis testing and confidence interval

The log return $X$ on a certain stock investment is an $N(\mu,\sigma^2)$ random variable. A financial analyst has claimed that the volatility $\sigma$ of the log return on this stock is less than ...
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630 views

How do I calculate the probability distribution of the percentage of a binary random variable?

I have an urn containing balls that are all either black or red. I'm interested in discovering the percentage of balls that are red. But I can only sample from the urn (without replacement), so the ...
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1answer
111 views

What is the distribution of the sum of n binary random variables with different probabilities and payoffs each?

Specifically, you can assume we have n random variables $X_i$ ($i \in \{1,2,3,\ldots,n\}$). Each $X_i$ has a probability $P_i$ to payoff $\mathrm{UP}_i$ and probability $Q_i=1-P_i$ to payoff ...
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81 views

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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4k 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
57 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
292 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
86 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
449 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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1answer
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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91 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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2answers
598 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
137 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
192 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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2answers
13k 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
698 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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245 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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303 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
812 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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122 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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54 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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98 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
857 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
343 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
783 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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1answer
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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
435 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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1answer
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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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
387 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
80 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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0answers
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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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1answer
156 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
67 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
107 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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125 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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363 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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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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2answers
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
253 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 ...