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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302 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
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0answers
5 views

Diffrence between chaotic and Orderd Equations/systems

what is the difference between chaotic and Ordered Equations ? if the first one means that the system behaviour is not predictable , why we just call it stochastic process , and using random ...
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3answers
23 views

Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $ \theta$ and variance $ a>0$. Let $ Y $ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
3
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1answer
20 views

Statistical Estimation Book Request

I am seeking a clear book for parameter estimation, estimation methods, properties of estimators, minimum variance estimators, asymptotic properties of estimators and interval estimation reducution.
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1answer
22 views

Scatterplot Confusion

I have just entered data into a scatterplot (by hand and by calc.), and for some reason the correlation is negative. This is weird due to the context of the problem: the x-axis is how long a swimmer ...
0
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1answer
34 views

Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
1
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1answer
457 views

How are the values in the Shapiro Wilk weight table calculated?

http://www.real-statistics.com/statistics-tables/shapiro-wilk-table/ I am trying to build a function that performs a Shapiro-Wilk test on a sample population, but I cannot figure out how the a values ...
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2answers
317 views

Confidence interval for Uniform($\theta$, $\theta + a$)

I am encountering a difficulty with the following task. Have I made a mistake, or is this an inherent flaw in the notion of confidence intervals? (Other such flaws exist.) Consider a random sample ...
5
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2answers
65 views

Update a regression on the fly?

Say I have 100 people each with a height, weight, and age. I make a regression that predicts age based on height and weight. Now, I would like to update that model when I meet someone new. I don't ...
4
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2answers
36 views

Determinant of Fisher information

In information geometry, the determinant of the Fisher information matrix is a natural volume form on a statistical manifold, so it has a nice geometrical interpretation. But what is it in ...
0
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1answer
220 views

UMP of a Beta($\theta,1$) distribution

I need to find the UMP of a BETA$(\theta,1)$ distribution. I know that the pdf of this problem is $$f(x;\theta)=\theta x^{\theta-1}=\theta e^{(\theta-1)\log{x}}$$ Thereby this family contains a MLR ...
0
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1answer
24 views

Risk function of binomial random variable

Suppose $X\sim Binomial(100,\theta)$, True estimator($\delta$) $= X/100$; $$R(\theta ,\delta) = E_{\theta}\left[\left(\theta - \frac{X}{100}\right)^2\right] = \theta(1-\theta)/100$$ I am ...
0
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1answer
234 views

Continuous random variables word problem

I'm having difficulty with a word problem on continuous random variables. Would you be able to help? Here it is: If you're n minutes early for an appointment, then you are charged n times 3 dollars, ...
3
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1answer
228 views

dynamic mean: measurement of randomly distributed events

Aim is to estimate an error on a stochastic event rate. I read out the event counter second-wise, every black $1$ is a counted event (new events over time, see the plot below). During the measurement ...
2
votes
1answer
19 views

Units of vaiance when variable is in %.

I have some confusion here. If some random variable is measured in some units, say $kg$ then clearly it's variance is measured in $kg^2$. But if the variable is dimensionless and measured say in ...
3
votes
3answers
81 views

How many ways can $10$ digits be written down so that no even digit is in its original position

If I have the numbers $0,1,2,3,4,5,6,7,8,9$ written down in that order, how many ways can the $10$ digits be written down so that no even digit is in its original position? It would seem that I can ...
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2answers
25 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
0
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1answer
43 views

History of Mathematical Formulas

I just wondered why in calculating something for example Variance we square the difference of the value and its Arithmetic Mean and do not take the absolute value of the difference? Are there books or ...
0
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1answer
130 views

find the average of a student [closed]

Five students in a closed room want to find average of their exam scores without revealing their personal score to each other. How can they do it? Hint: they can write something on paper and pass it, ...
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1answer
22 views

Find joint distribution function in region

I can't for the life of me figure this one out, I am stuck on part (c) ... I have this as my starting point ? $$ \frac{45}{304}\int_0^x\int_{2-x}^2 u^2v^2\,\mathrm{du} \mathrm{dv} $$ Here is my ...
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0answers
24 views

SBC score clarification

For the SBC test, it says that this is more favourable as compared to the Anderson darling, kolomogorov Smirnoff test and the chi-squared tests because it doesn't have the fault that, when n increases ...
1
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1answer
25 views

Prove that mean square error equals expected conditional variance

I'm a first year grad student in Statistics. The book I'm using mentioned conditional variance, and I wanted to read up more about it. I dove down the google rabbit hole and found this website. I read ...
0
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0answers
16 views

What does a presentation on block design and Latin squares consist of?

I read the wikipedia pages of both and I just cannot understand these two concepts. I have a presentation on both of these topics next week and I need some headway on both of these topics.
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1answer
28 views

When does Cramer-Rao Lower Bound fail to hold?

When does Cramer-Rao Lower Bound fail to hold? I have computed the bound, and still one of my estimators has less variance below the bound. Why does this happen? Should not this bound hold always?
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0answers
15 views

A question regarding 90 ball bingo [on hold]

What is the average number of calls necessary to daub a number on a ninety ball bingo ticket?
2
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2answers
73 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
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0answers
18 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
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1answer
17 views

What are the chances of a sample distribution, given that I know the larger population's distribution?

I know that the larger population, of size m, follows a normal distribution. I have some sample that is size n, and follows its own particular distribution, which is not normal. Given my sample sizes, ...
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1answer
105 views

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 ...
1
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1answer
229 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
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0answers
31 views

Economics, Statistics

Now suppose I am trying to forecast using historical data and using random walk method, In the formula $ y_t = y_{t-1} + u_t $, how can I find the $u_t$ form the model ?
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2answers
250 views

Statistics about unbiased estimator with an unknown parameter

Way too hard questions for me. Could anyone give me some clues or way to solve?
0
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3answers
94 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?
3
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4answers
126 views

Question on the Condorcet Ranking (Schulze Method) Algorithm

I've been doing some research on ranking algorithms, and I've read theresearch with interest. The aspects of the Schulze Algorithm that appeals to me is that respondents do not have to rank all ...
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1answer
219 views

Statistics Question of unbiased estimator

Could anyone give me a hint? or way to solve?
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2answers
39 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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0answers
23 views

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. ...
2
votes
1answer
520 views

How to calculate correlation between poll statistics and one's answers?

I have statistics of $100$ questions which can be answered either "yes" or "no": 1) $63.3 - 36.7$ ($63.3\%$ respondents answered "yes" and $36.7\%$ answered "no") 2) $30.1 - 69.9$ ... 100) $88.0 - ...
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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 ...
9
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6answers
4k views

Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
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0answers
20 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 ...
0
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0answers
14 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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0answers
31 views

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 ...
2
votes
1answer
337 views

Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to ...
1
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1answer
31 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 ...
1
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1answer
224 views

Probability of a slot having exactly $K$ elements

From this question asked in an interview: Consider a hash table with $M$ slots. Suppose hash value is uniformly distributed between $1$ to $M$. Suppose we put $N$ keys into this $M$-slotted ...
0
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4answers
35 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 ...
2
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1answer
268 views

Clayton copula and Kendall's tau

I'm currently preparing for an exam in Risk Management (mathematics) by doing exercises from old exams. One of these exercises proved to be too difficult because of the following: Given Kendall's tau ...
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1answer
21 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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2answers
81 views

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} ...