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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How to convert a problem to a stars and bars problem?

Continued question from here. With certain questions I have $x_i$ being constrained by various different inequalities, I want to know how to remove these from the problem, to bring me back to a ...
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2answers
32 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
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Math issue implementing an invoice API [on hold]

Okay, so, I have $2$ separate systems: An invoice record database on an external site, I do not have access to the code here. An prestashop e-commerce installation, where i am developing a plugin. ...
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1answer
11 views

discrete variables in fraction

I am looking for examples of discrete variable whose value is in fraction ONLY. I am told there are very few such examples. I am not looking for variables with mixture of both whole numbers and ...
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8 views

Jeffrey's Prior for Bivariate Lognormal

Exactly what the question says, I'm working on code for an MCMC simulation and need to set some uninformative or weakly informative priors. I haven't been able to find the prior for the sigma ...
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11 views

Limiting distribution of loss random variable?

I'm going to try to make the notation not actuarial-specific, but for those with a background in actuarial science, this relates to exam MLC. Suppose I have random variables $X_{i} \geq 0$ such that ...
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4answers
30 views

Probability: Linear Seating Arrangement

Okay, I'm new at probability and statistics, so please try to answer this as thoroughly as possible and explain why you did everything, from using a specific number to why using factorials and ...
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1answer
20 views

Expected length of a random vector

I meet a basic definition about the expected length of a random vector when reading a paper: What is "expected length" How to roughly derive both equations (yellow part) (Is that Gamma ...
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3answers
79 views

How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
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Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
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1answer
18 views

Compound Interest - Initial Amount Borrowed [on hold]

Could someone explain how to calculate this question? Two loans of an equal amount were taken out at $10\%$ interest per annum for $3$ years and $4$ years respectively. The difference in the ...
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1answer
14 views

Avg # of Rectangle Intersections in 2D Field

So imagine I have a large 2D field. Thousands of small rectangles overlay the field. The field is much larger than the rectangles. The rectangles are placed randomly in the field such that they may or ...
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23 views

help check the derivation of joint pdf of the sample covariance with Hermitian circulant structure

The question comes from evaluating the covariance matrix and its moments of samples received by a circular array. the covariance matrix is proved to have Hermitian circulant structure. Suppose the ...
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29 views

Estimators of the binomial distribution

This is a follow-up of this previous question and elaborates on the answer I received there. $\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}\def\a{\alpha}X_1,X_2,\dots, X_n$ are a random sample from ...
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1answer
26 views

How to predict next number from a given set of measurement data?

I have to do some experiment and measure it on a specific time 0, 3, 6, 9, 12, 18, 24, 36 months $$ \begin{array}{l|c|c|c|c|c|c} \text{Month} &ID1 & ID2 & ID3 & ID4 & ...
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1answer
28 views

Need my data to fit ANOVA…

I am doing some research on constructed wetlands. I have four wetland as follows. Wetland A - Gravels Wetland B - Gravels + Plant Wetland C - Biochar + Plant Wetland D - Biochar + Gravels + Plants ...
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1answer
25 views

How to estimate the standard deviation in this normal distribution?

There is this simple looking basic statistics question that asks to estimate its mean and standard deviation. I have some doubts and just want to make sure whether my working is correct. For part ...
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1answer
17 views

Standard deviation in normal distribution

A manufacturer uses a machine to make metal rods.The diameter of the rods follow a normal distribution with a mean of 1cm and a standard deviation of 0.02cm If the standard deviation of the diameters ...
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10 views

find the mean of a collection of the solutions of given linear systems

Suppose one has a group of $k\times k$ square matrices $A_i, i = 1,2..n$, another $k$-vector $b$ is also given. I want to exam the mean of the group of vectors $x_i$ over $i = 1,2...n$ where $A_i * ...
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1answer
19 views

Simplify $\dfrac {\partial }{\partial b}\left( \sigma ^{2}\right) =0$ and $\dfrac {\partial }{\partial m}\left( \sigma ^{2}\right) =0$

Given $\sigma ^{2}=\sum ^{N}_{i=1}\left[ y_{i}-\left( mx_{i}+b\right) \right] ^{2}$ How to simplify $\dfrac {\partial }{\partial b}\left( \sigma ^{2}\right) =0$ and $\dfrac {\partial }{\partial ...
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19 views

Negative integer factorial in Koziol's combinatorial approach

I' m trying implement Unbiased split selection from this document: http://www.stat.uni-muenchen.de/sfb386/papers/dsp/paper464.pdf I have problem on 14 page. I have list of $$ \binom{i_k}{j_k} $$ $$ ...
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1answer
12 views

Finding the percentile of a value given other percentiles

I am using someone else's research and only have part of the data. I want to find the percentile of a value, such as 350.2, based upon the data given. The following is known: mean = 308.2 The ...
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2answers
13 views

How to prove a given statistical test has the greatest power

It is very conventional in evaluating the null hypothesis to consider the distribution of the mean of a sample of some size if the null hypothesis were true and to compare the mean of your own sample ...
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0answers
11 views

Comparision between Hamming distance and cosine similarities?

I want to check the similarities between binary vectors of different length and I am using cosine similarities and hamming distances for calculations. These are of length 1000 elements(0 and 1). ...
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1answer
20 views

Median of grouped data

part (ii) Median = $Lm +(n/2-cf)/fm*h$ = $171+(20-16)/12*3$ = $172$ am i correct?
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1answer
28 views

Statistical Inference - Finite Population Correction

I am studying for an exam in Quantitative Analysis and I am looking through past exam papers (here's a link the exercise I'm referring to: http://i62.tinypic.com/2cz1nqh.jpg). I understand all of the ...
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1answer
27 views

Equivalence of two formulas for variance and covariance

I know two formulas for variance: $$\operatorname{variance}(f) = \operatorname{expectation}((f(x) - \operatorname{expectation}(f^2(x)) \\ = \operatorname{expectation}(f(x)^2) - ...
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1answer
25 views

How to derive “Pooled Sample Variance”?

Let $s_p^2 = bs_1^2 + (1-b)s_2^2$, this can be an unbiased estimator of population variance, provided we find the correct value for $b$; in particular, $s_p^2 = \frac{(n1-1)s_1^2 + ...
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1answer
37 views

Statistics Probability Density Functions with Mutliple Features (Multivariate Normal Distribution)

I'm looking for a beginner-friendly explanation on how this Probability Density function works when dealing with mutliple features and what the variables and terms mean in detail. I'm seriously ...
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1answer
30 views

A Bayesian estimate for rank

I am not sure how to put this succinctly but here is the problem. Consider a random variable $X$ which can take a finite number of non-negative values. It takes each value with the same probability. ...
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3answers
58 views

Derivation of standard error of mean

I was going through this wikipedia article on standard error. I could not understand the crucial step here. It goes like this: This formula may be derived from what we know about the variance of a ...
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2answers
60 views

Does an exponential model fit my data?

I am measuring accumulation of a fluorescent-tagged protein at a particular location within a cell over time. In previous experiments that I have performed, I see a standard exponential distribution ...
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1answer
27 views

time series similarity heuristic

What is a good heuristic for time series similarity. I'm ok with false positives bc I will then pass the results of the heuristic to an actual series similarity algo. Basically I want the heuristic to ...
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1answer
59 views

Math required for medical statistics

I have never been good with Math. I talk to many of my medical colleagues and it seems to me that most of them have a poor understanding of statistics. Many of them claim to understand it but actually ...
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1answer
21 views

How to calculate lower & upper quartiles?

I'm sure this has been asked many times before but it's confusing me a lot so hopefully someone can help! I am given this data set: 0.28 0.30 0.42 0.59 0.71 1.67 1.82 2.39 4.71 4.79 4.89 5.00 5.00 ...
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2answers
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How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
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1answer
28 views

Why the standard deviation approximated by the range rule of thumb much different from the standard deviation calculated directly from sample data?

What makes the standard deviation approximated by the range rule of thumb much different from the standard deviation calculated directly from sample data? For example, if the approximate standard ...
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33 views

Outliers in a Normal Distribution

Im doing AP Stat. in High School level. Here is a question i am stumped on because i feel like it is maybe a threory or law or something that i just never learned. However it DOES ask to show my ...
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1answer
23 views

Finding moment estimator

I am required for a homework question to find moment and maximum likelihood estimators, and in class we have only looked at finding MLEs and using already known moment estimators, hence I have no idea ...
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1answer
25 views

Find the Distribution of an Order Statistic

I am working on a practice prelim question. It states: Let $Y_1 < Y_2 < Y_3 < Y_4$ denote the order statistics of a random sample of size 4 from a distribution which is uniform on ...
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1answer
28 views

Interpretation of a statistical formula involving the ratios of sample and population

Kind of an odd question, but, is this a standard equation for stats? I can't figure out for the life of me what it looks like. $\left(\frac{\text{Sample Of A}}{\text{Population Of ...
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1answer
20 views

Calculating expectation and using the delta method

Let $\hat{p}$ be the proportion of successes in $n$ independent Bernoulli trials each having probability $p$ of success. (a) Compute the expectation of $\hat{p} (1-\hat{p})$. Isn't $P(\hat{p} = k) = ...
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1answer
9 views

Derivation of Mode of grouped data

A formula to calculate the mode for grouped data's is given in my text book: Mode = $l +$ $ (f_1 - f_0) \over (2f_1 - f_0 - f_2) $ $ * \ h $ Where, $l = $ lower limit of the modal class, ...
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1answer
34 views

A Bayesian estimate of the bias of a coin

Consider a coin with unknown probability $p$ of landing on head. I will toss the coin and stop as soon as I get a head. Say this is after $n$ tosses. If my prior belief for $p$ was uniform on ...
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1answer
25 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
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1answer
27 views

Estimator with smallest variance, Lagrange multiplier

I have a question and I think I'm supposed to use the Lagrange multiplier although I haven't been taught it, so I'm not sure if I can use it or not. The question is: Suppose that $X_i$ has mean $\mu$ ...
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1answer
9 views

Showing deviation of data on a fixed scale

I'm creating a chart in D3 which shows the Mode from a large (1mil +) dataset; and colors it according to the standard deviation of the dataset. But the issue is that I need to map the range (Red ...
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1answer
55 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
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15 views

Expectation Maximization Algorithm for Gaussian Mixture Model

Can we use the Expectation Maximization algorithm for estimation of Gaussian Mixture Model with full covariance matrices? If yes then can you please give me a reference paper? So far all the machine ...
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4answers
82 views

Estimate bias of a coin

Consider a coin with probability $p$ of landing on head. You can estimate the prob by tossing it lots of times and looking at the proportion of heads one gets. In my problem I just want to know if ...