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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Features of a Set Given Same Value for Maximum and Average

What else, if anything, can be said if the maximum of a set is equal to the average value in the set? If the average is equal to both the maximum and minimum then all the values are the same, but can ...
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3answers
34 views

Find the mean and median help!

I feel like there isn't much information, but the answer for the mean is 79. The median is said to not be able to be found. Can someone explain this? Thank you!
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85 views

Difference between Kendall tau distance and Kendall tau rank correlation coefficient

What is the difference between Kendall tau distance and Kendall tau rank correlation coefficient? Which one I should use to calculate how similar two arbitrary permutations of ranks are? Suppose, we ...
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145 views

Probability Question involving Probability Mass Function/Random Variables

Problem: When a paging system transmits a message, the probability that it will be received correctly by the appropriate pager is p. To ensure that the message is correctly received at least once, the ...
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89 views

Derivative of an exponentially weighted moving average

It has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however, ...
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67 views

Difference between ANOVA and Pearson Chi-squared test

To test the equal of variance the non categorical variable using ANOVA test. As for the data with categorical variables using Pearson's Chi-squared. In this case, why the ANOVA test can not be used ...
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22 views

A simple plot of a list in R

This is a very easy question, I guess. But unfortunately I struggle to solve it in an easy way: I have a list in R. I would like to plot the entries of each column vertically and label the x-axis to ...
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34 views

Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
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223 views

Find a 10% likelihood interval

The function is: (n choose x)$[(1-y)^{k}]^{x}[1-(1-y)^{k}]^{n-x}$ Suppose n = 100, k = 10, x = 89 I found the maximum likelihood of y-hat to be 0.0116 Now I need to find a 10% likelihood interval. ...
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25 views

How could you find the probability that the estimator is within 0.03 of the mean?

p = fraction of large population that smokes n = sample size y = # in sample that smoke The maximum likelihood estimate of p is p-hat = y/n Consider the random variable Y and estimator F = Y/n ...
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22 views

Density of k-th order statistic

I know that the k-th order statistic density for the uniform distribution is given by: $f_{(k)}(y) = \frac{N!}{(N-k)!(k-1)!} t^{k-1}(1-t)^{N-k}$. This is the density if the $N$ samples are sorted in ...
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12 views

Generating a given length sample and skewness whose normality is verified by one normality test but not by an other

I just want to generate 1 sample of length$=n>30$, |skewness$=S|<0.3$ and for which normality is not rejected by Shapiro wilk test of normality but rejected by Anderson darling test of ...
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40 views

two correlated process

I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out. Assume that there are two ...
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42 views

Multivariate random variable. Density function

Assume that the random variable $X$ take values in the closed unir hypercube $[0,1]^{d}$ for some $d\geq 2$ fixed. Put $X:=(x_{1},\ldots,x_{d})$. Given $m\in \mathbb{N}$, it is not very hard to show ...
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31 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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115 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
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26 views

Calculating Variance

Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a population whose distribution is normal with mean $\mu$ and variance $\sigma^2$. Consider the statistics $\displaystyle T_1 = \frac{X_1 − X_2 ...
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30 views

Notation related to Markov kernels

We wish to jointly construct two copies $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ of a Markov chain on general state space, s.t. for $n=1,2,...$ $\mathcal{L}(X_{n+1}|X_n) = ...
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34 views

Parameter estimation of Lorenz system (nonlinear dynamical system)

My problem is as follows. I have to estimate parameters of Lorenz system using given data. Lorenz system is described by following system of ODEs: $$ \frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = ...
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14 views

Measuring availability of a service

I have a task to measure availability of some IT-components. The problem is that I need to create a equation for measuring this. For now the availability is going to be measured through life-cycle ...
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60 views

R squared (Proportion of variance explained) in terms of conditional variance?

My question concerns a comparison between 2 models in terms of proportion of variance explained. Let $y_{t+1}$ denotes the variable I want to explain or predict and $\mathcal{F}_t$ the information ...
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39 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
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35 views

Law of large numbers for U-Statistics with varying kernel

Let $U_n=\binom{n}{2}^{-1}\sum_{1\leq i<j\leq n} H_n(X_i,X_j)$ be an $U$-Statistic of order $2$ and with kernel $H_n$ depending on the sample size $n$. I wonder if there exists a Strong (or Weak) ...
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26 views

What statistical model can be used to find how following variables relate?

I want to find the relationship between a disease incidence in a region and a number of other environmental factors such as temperature, elevation etc. I have tables containing this data for a ...
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23 views

Question on rounding off rule

There is a specific rule(Banker's Rule I think) for rounding of numbers that end in 5. The rule is that we add 1 to the preceding digit of it's odd but keep it as it is if it's even. It's always ...
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30 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
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1answer
46 views

What is the distribution of the product between a continuous (Exponential) and a discrete (Bernoulli) random variables?

If you have $X$ distributed as a $\mathrm{Bernoulli}(p)$ and $Y$ as a $\mathrm{Exponential}(\lambda)$ find $Z=XY$. I tried doing it with the MFG i.e. ...
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1answer
33 views

how to check the variables which underwent the maximum change in a time course data?

I have time series data of many variables (these data for these variables are coming from a high-throughput experimental studies) The time series looks like $x \to x_1, x_2, x_3..$ which is the value ...
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38 views

Question about Logistic Regression - 2

How should I tell the difference between those two formulas in the circles below. I am studying logistic regression and I have faced two different formulas from two different documents. I don't know ...
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32 views

Distribution, parameters, expectations?

You regularly attend baseball games. At these games, you have two methods for receiving free souvenirs: either by catching a foul ball or by catching a t-shirt thrown by the team mascot. Assume that ...
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53 views

Optimal strategy in a simple card game

So some friends and I are playing this simple card game. We choose 15 random cards from a normal deck of card and lay them on a table in a pyramid shape. 5 cards in the bottom row, 4 in the next row ...
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15 views

Singularity of marginal and joint distributions

can anybody give me an example of random vector (X, Y) such that its joint distribution is orthogonal (singular) to the product of marginal distributions of random variables X and Y? Thank you very ...
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31 views

Normal distribution of t

I have a population with 10000 individual, i took 10 to my samples. The values that i have inside $\bar{X}$ $$70.22502193, 70.26042017, 70.28977621, 70.2905717, 70.30113496, 70.32304453, ...
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29 views

Modifying a Density Function

Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria: $$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad ...
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21 views

PDF, probability, variance, statis

Let X be normal with mean 1 and variance 4. LetY =2X+3. Find P(Y ≥ 0) I've solved E(Y) = 5 and Var(Y)=16. How do I apply this solve $P(Y\geq 0)$
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3answers
29 views

How can I do step-by-step calculations for the three equations to be equivalent to each other?

I am working on logistic regression, and have faced with the three equations below. I am told they are all equivalent. Why, how can I do the step by step calculations for them to be equivalent? ...
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1answer
18 views

Understanding details of PCA - variance

I am currently reading through Principal Component Analysis, Second Edition and came to small paragraph that I do not fully understand on page 5 (Section 1.1): "To derive the form of the PCs, ...
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1answer
34 views

Question about Logistic Regression - 5

Can somebody give me a clear explanation about logistic regression in bold below? Logistic regression can be binomial or multinomial. Binomial or binary logistic regression deals with situations in ...
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70 views

World cup birthday paradox for two pairs

The BBC report on the world cup squads The birthday paradox at the World Cup shows the 50:50 prediction for there being 16 teams out of the 32 that meet the pair birthday criteria. However my ...
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1answer
72 views

Hypothesis testing: Test statistic, P-value and significance levels

A manufacturer claims his light bulbs have a mean life $μ = 1800$ hours. A consumer group tested a random sample of $n = 250$ bulbs and found them to have a sample mean life $\bar{x} = 1790$ hours ...
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1answer
82 views

estimation problem for two-parameter weibull distribution

Suppose the two-parameter Weibull distribution is given by the pdf $$ f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\}, $$ where $x,a,b>0$. I am ...
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1answer
352 views

Statistic about Actual vs. Projected

I am trying to show how much a certain thing has exceeded or fallen below its expectation. How do I calculate this? I forgot if this statistic is called Percent Difference or something else, I ...
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2answers
149 views

convert continuous random variable to a discrete one for the given exponential distribution

I understand that the following question requires converting continuous r.v. to discrete r.v. But How can we get a PMF from the CDF of continuous distribution? It involves dividing continuous values ...
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1answer
32 views

Maximum likelihood estimator transformed parameter

I don't get the gist of b). What is it that we are in fact calculating here? I don't get why we can just plug in the rearranged formula.
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26 views

What is The Basis for The Similarity Of Distance And Standard Deviation?

The common way to think about distance is Minkowski difference with the special case $r=2$. The definition for standard deviation is $ \sqrt{\frac{1}{n-1} \sum (x_i - x)^2}$. Both of these definitions ...
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1answer
29 views

Understand the English paragraph on association rule.

I am currently studying Association Rule Pattern Mining. I am reading the explanation on wikipedia about it. Somehow, I feel like I have a problem in understanding the paragraph below. Can somebody ...
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15 views

Question about the poly-Euler preprint

I'm looking over the preprint on poly-Euler numbers, and I think there might be a typographical error on the third line because $j=0$ leads to a discontinuity. Could somebody explain the substation ...
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1answer
22 views

When using the geometric mean to compare data with different ranges, how do you include a binary piece of data?

For instance, if you were comparing 10 companies on 3 categories, with the first category having a range of 1-5, the second having a range of 1-100, and the third being either 0 or 1. All three ...
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32 views

Multiple correlation

My question is very elementary. I need to implement a software that computes the multiple correlation of a set of datas that I have. In order to do that, I need would like to find the generic formula ...
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1answer
30 views

Best way to handle the ratio which cannot be represented as floating point numbers.

I need to calculate the ratio of the form: $s=\sum_1^3q_i$,$\quad$ $p_i=\frac{q_i}{\sum_1^3q_i}$, where $q_i >0$. One problem is that $q_i$ are too small that they can not represented as ...