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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31 views

What does MSSL sampling refers to?

In a document I came across this sentence "calculation of MSSL using fixed and dynamic methods". The document itself is about sampling techniques and no other explanation is given there. I tried to ...
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144 views

Simple probability with marbles

I have completely forgotten my statistics knowledge, and I have now found myself in dire need of this without having access to my books. While I'm pursuing another context, I would like to ask my ...
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276 views

Statistics and confidence - intervals

An account on server A is more expensive than an account on server B. However, server A is faster. To see whether it's optimal to go with the faster but more expensive server, a manager needs to ...
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36 views

$X$ random uniform variable, what is the cumulative distribution of $|X|$?

Let $X$ be a random uniform distribution on $[-2,1]$. What is the cumulative distribution of $|X|$, i.e. $G(x) = p(|X| \leq x)$ ? The cumulative distribution of $X$ is $F(x) = \dfrac{x+2}{3} $. ...
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186 views

Random variable with density proportional to a function and finite in some points

Let $X$ be a random variable on $[-1,3]$ with density $f(x) = k x^2$ (with $k \in \mathbb{R} $ to be determined) on $[-1,3]$ apart from some points s.t. $p(X=-1) = p(X=3) = \dfrac{1}{4} $ and $p(X=0) ...
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36 views

How to prove statistical hypothesis?

I developed a caching method. I took 100 experiments and got that hit ratio is not less than 75%. Now, I want to prove that my method with some probability gives hit ratio not less than 75%. How ...
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61 views

Questions Regarding Linear Regression

Are the slope and intercept of a simple linear regression model always normally distributed? Is there ever a difference between the distribution of the estimated slope and intercept and the actual ...
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597 views

Cramer-Rao lower bound for the variance of the unbiased estimator of $\tau(\theta)$

Given the pdf $f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$ ; $-\inf < x<\inf$, $-\inf < \theta<\inf$ Show that the Cramer-Rao lower bound is 2/n where n is the sample size.
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209 views

Trying to understand an example where unbiased estimators don't exist

I am new to statistics especially in the topic of estimators and sufficient statistic. I am reading a note which says "unbiasedness is a desirable (but not necessary) property of a good estimator". ...
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154 views

Poisson point process convergence

Let Π be a Poisson point process on [0,∞) with intensity measure $\mu$. Assume $μ([0,t])<∞$ for all $t<∞$ and $μ([0,∞))=∞$. Also assume $μ({x})=0$ for all x. Prove ...
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228 views

Is pi lying on the ground, and on TV?

Consider the leaves from a bunch of trees in a terraced plaza in the Autumn. It may well happen that the tiles of the terrace are squares whose length easily exceeds the length of the stem of the ...
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45 views

Does the 80-20 Rule apply to the Buffon process?

Whether the needle crosses a line depends greatly on the angle the needle makes with the line, 90 degrees being of course the most favorable for a line-crossing. Does the 80-20 Rule apply to these ...
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31 views

Does the Law of Factor Sparsity apply to factorization?

Does the 80-20 Rule (also known as the Law of Factor Sparsity, the Law of the Vital Few, and the Pareto Principle) apply to factorization? That is, if x is a large positive integer, then are about 20% ...
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2answers
574 views

Linear Regression - Proof Sum Adds to Zero

In linear regression, why is $\sum(X_{i} - \mu_{x})$ = $0$? I understand that for ($\sum$ $Y_{i}$ minus the fitted value of Y) = $\sum$ $e_{i}$ this is true but why is this other fact true?
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1answer
115 views

Total probabilities of being admitted to any university

Let's provide an hypothetical situation in which a student applies to 10 different universities whose number of applicants, admissions and admission rate you can see in the table below. ...
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1answer
273 views

Calculating PACF of $ARMA(p,q)$

I am struggling with the calculation of PACF (partial autocorrelated functions) in $ARMA(p,q)$. The lecture notes I have gives an example of calculating PACF of $AR(1)$. For ...
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328 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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99 views

correlation of product with its normally distributed factors

If x and y are normally dist. with standard deviation of 10%, and they are independent, then their product X.Y is 71% correlated with Y (or X). I can show this empirically, but how to I prove it in ...
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435 views

Uniform distribution joint $\to$ marginal

Let vector $(X,Y)$ have a uniform distribution on the set $N = \{ (x,y): x<1,y<1,1<x+y\}$. Determine distribution $X-Y$. So far I've thought of this: \begin{align} P[X | Y=y] &\sim ...
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1answer
223 views

What makes the Var(x)+Var(y)=var(x+y) property important?

What makes the Var(x)+Var(y)=var(x+y) property important? It was taught in my statistics class
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1answer
49 views

help me with this regarding hypothesis using chi square distribution

The rope used in a lift produced by a certain manufacturer is known to have a mean tensile breaking strength of 1700 kg and standard deviation 10.5kg. A new component is added to the material which ...
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34 views

this is regarding exponentials distribution

In an office building, the lift breaks down randomly at a mean rate of 3 times per week. The random variable X represents the time in days between successive lift breakdowns. (i) Calculate the ...
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2k views

Exponent p-value generated in Excel

Excel gave me a p-value of 1.44909E-09 Notice is does not say .09 but 09 This is confusing me, I am trying to analyze my data but am stuck at this point. If it were E-9 it could be ...
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2answers
148 views

What is the purpose to define central moment?

What is the purpose to define central moment? I searched the google and all i could find is bunch of properties
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1answer
122 views

Error Propagation in Successive Least Square Adjustment

I have a certain problem in surveying in which I'm trying to do some error analysis. I'll layout the problem first. My current goal is to evaluate the variance-covariance matrix for the position of ...
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4answers
8k views

How to calculate R-square from adjusted r-square, n, and p?

Let $\bar{R}^2$ denote the adjusted coefficient of determination. I have $\bar{R}^2 = 0.9199$ with 15 cases. Now I am trying to find $R^2$ given the results below. I found the formula for $R^2$ but ...
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422 views

Acceptance sampling schemes for binomial distribution

Two acceptance sampling schemes, A and B, are proposed for deciding whether or not to accept a large batch of items from a production process in which 5% of the items produced are defective. Scheme A: ...
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1answer
133 views

Probability of rolling three dice without getting a 6 in 8 rolls

I know the propability of rolling three dice without getting a $6$ in $1$ roll is $91/216$, but what about rolling it in $8$ rolls consecutively with three dices?
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60 views

Basic question on the transformation of Exponential distribution.

Why central moments coincide for random variables $V\sim E(a,h)$ and $Y\sim E(h)$ where a=location parameter h= scale parameter.
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1k views

Covariance of order statistics (uniform case)

Let $X_1, ..., X_n$ be uniformly distributed on $[0,1]$ and $X_{(1)}, ..., X_{(n)}$ the corresponding order statistic. I want to calculate $Cov(X_{(j)}, X_{(k)})$ for $j, k \in \{1, ..., n\}$. The ...
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1answer
37 views

How can I measure variance efficiently?

I have bunch of values, for example {1,2,3,4}. I need to measure variance in a very efficient way. On wikipedia variance is defined as sum of squared differences between the data examples and the ...
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3answers
424 views

Statistics Question

This is probably super simple to most of you on here, but I was chatted by a friend earlier with a question. It reads just like this: ...
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106 views

find $\theta_{MLE}$ for a function

For $$ f(x;\theta)=(\theta+1)x^{-\theta-2} $$ find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$. My work so far: $$ \begin{align} \prod_{i=1}^n ...
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299 views

Simple Linear Regression Question

Let $Y_{i} = \beta_{0} + \beta_{1}X_{i} + \epsilon_{i}$ be a simple linear regression model with independent errors and iid normal distribution. If $X_{i}$ are fixed what is the distribution of ...
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1answer
321 views

Ratio of PDF to complementary CDF

Let $f(x)$ be a probability density function, and $F(x)$ be the cumulative distribution function of $f(x)$. $$F(x) = \int_{-\infty}^{x}f(u)du$$ Then intuitively, what does the following ratio ...
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1answer
776 views

Find the limiting distribution

Find the limiting distribution for $n\rightarrow \infty \text{ of} \prod\limits^n_{i=1}X_i$. Given is that $f(x)=\frac{1}{2x\sqrt{2\pi}}e^{-\frac{1}{8}(\ln x-\theta)^2}, x\geq 0$.
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1answer
76 views

This is regarding Chi square test

A chi square test is conducted to check whether a person's ability in Mathematics has an impact on his/her interest in Statistic. The test statistic is 13.277 under the tested null hypothesis. write a ...
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41 views

Calculating the probabilities of different lengths of repetitions of numbers of length 6

This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4 except now I'm having problem with numbers of length 6. ...
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1answer
54 views

Distribution for Response Times

I have samples from a response time population for a web transaction. I want to be able to use them to describe a distribution for the population but don't know a proper one to use. I have shied away ...
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1answer
126 views

Wealth indicator function for bidder agent logic

I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others). Given: Total number of competitors $n$ Amount of all ...
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1answer
140 views

Find the Cramer-Rao bound for an unbiased estimator of $b^2$

$X$ is a RV with pdf $f(x,b) = \frac{x}{b^2} \exp \{-\frac{x^2}{2b^2} \}$ I've got two different estimates: $\hat{b^2} =\frac{2}{\pi} (\frac{1}{n} \sum_{i=1}^n X_i)^2 $ using MME, and $\hat{b^2} = ...
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2answers
212 views

How do I divide a set of data samples which follow a logarithmic distribution?

I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
2
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1answer
76 views

Continuous random variable question

$ X $ is a non-negative continuous random variable with density function $f$ and distribution function $F$. Use integration by parts to show that $ \int_0^{\infty} ( 1- F(x)) dx = ...
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2answers
82 views

Check for sufficiency

We have the function $$f(x) = \frac{1}{2x\sqrt{2\pi}}e^{-\frac{(\ln x - \theta)^2}{8}}$$ for the I.I.D (identically independent distributed) sample $X_1,\dots, X_n$. I have to show that ...
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0answers
30 views

Time Series: Whether a proces is invertible or not ?

If the roots of a characteristic polynomial of a MA(2) process are on the unit circle, i.e. z = 1, is the process still invertible?
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2answers
78 views

expected value of random variables

Take two random variables $X=a+bX_0$ and $Y=c+dY_0$, and define $T=X-Y=\mu+\sigma Z$ where $\mu$ is the mean of $T$, $\sigma$ its standard deviation and $Z$ is a standardized random variable with mean ...
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1answer
45 views

Range of the distribution of $(1-X)$ when $X$ follows Beta distribution as $X\sim beta(p,q)$

if $X$ follows beta distribution with parameter $p$ and $q$ where $p>0\quad , q>0$ then $1-X$ follows beta distribution with parameters $q$ and $p$, that is if $X\sim beta(p,q)$ then ...
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1answer
24 views

Regular Conditional Bayesian Experiment

In "Elements of Bayesian Statistics" (1990), Florens, Mouchart and Rolin describe two basic forms of reduction of a Bayesian experiment: Marginalization and Conditioning (Ch. 1). I don't understand ...
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1answer
368 views

Showing uniform convergence in probability

Suppose you want to show $sup_{x\in D}|f_n(x)|\to_p 0$, for $n\to \infty$, where $D\subset \mathbb R$ is a compact interval, $f$ is continuous depending on one or more random variables, and $\to_p$ ...
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2answers
366 views

question on five number summary & quantile.

i know that in five number summary : 25% of a data set lies between Min & 1st quartile. 50% of a data set lies between Min & 2nd quartile, that is, Median. 75% of a data set lies between ...