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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Does “Cook's distance” tell us the outlier?

How many way to find the outlier? For cook's distance, which level is the cut off of outlier?
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109 views

Halloween candies!

Children go trick-or-treating in three mathematicians' apartments. In MathA's apartment, a child will roll a die and the number of candies the child receives will be the same as the outcome of the ...
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85 views

Theoretical justification for Cochran's rule in the $\chi^2$ test

There is an empirical rule performing a $\chi^2$ test for goodness of fit: the expected frequencies have all to be greater then or equal to 5. Does someone knows why? Is there any proof?
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102 views

Who established the word “ Degree of freedom ” in statistics?

I wonder who is the first one that established and applied the word : "degree of freedom" in statistics? Why he/she need degree of freedom in the calculation of many statistical values?
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1answer
287 views

Halloween candy picking probability!

Two persons, A and B are each picking up one piece of candy from a basket consisting 100 Kit-Kats, 200 Almond Joys, 300 Whoppers and 400 Skittles. A likes Skittles; B likes Almond Joys. What is the ...
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1answer
100 views

limit of an integral of a copula density function

let's say I have a copula density function which I denote as $c(x,y)$. $X$ and $Y$ are uniformly distributed RVs. I am curious if the following limit exists: $\lim_{u\rightarrow 1^{-}} \int_0^u ...
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334 views

Calculating Probabilities of a Random Variable Function

The problem is: A mail-order computer business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying ...
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1answer
229 views

trying to verify pdf for distance between normally distributed points

Math people: I am trying to find the probability density function for the distance between two points in $\mathbb{R}^3$ selected independently according to the Gaussian pdf $F(\mathbf{z}) = ...
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1answer
527 views

Sufficiency and UMVUE for Poisson distribution

I need to show that $\hat\lambda = \bar X$ is a sufficient estimator for a Poisson distribution iid $X_1...X_n$, show that $\hat\lambda$ is the UMVUE for $\lambda$ and that $\hat\lambda$ is a ...
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2k views

How do you calculate IQR (interquartile range)?

I have the following data (ordered): $$0, 1, 1, 2, 3, 4, 4, 7, 9, 23.$$ As far as I know, $Q_1 \text{(median of the upper half)} = 1$; $Q_3 \text{(median of the lower half)} = 7$; Therefore, ...
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88 views

Can someone help me with this formula?

I'm writing a software algorithm at the moment which compares survey answers. Questions have $5$ possible answers, and a respondent could choose between 1 and 5 answers. What I'd like to do, for ...
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1answer
163 views

Bandwidth selection for kernel density estimation, using a Weibull kernel

Let $\{s_1,\ldots,s_N\}$ be a collection of N samples. I have performed the kernel density estimation using the classical form: $$ \hat{f}(x) = \frac{1}{Nh}\sum_{i=1}^N K\left(\frac{x-s_i}{h}\right) ...
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1answer
133 views

Zipfs law and LogNormal distributions

If a particular dataset has a lognormal distribution, will it also follow Zipf's law when the items are ranked? That is, say I have a set of populations of a random sampling of cities (assumed to be ...
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2answers
1k views

Show that $\hat\theta=\frac{2 \bar Y- 1}{1- \bar Y}$ is a consistent estimator for $\theta$

Let $Y_1,Y_2,...,Y_n$ denote a random sample from the probability density function $$f(y| \theta)= \begin{cases} ( \theta +1)y^{ \theta}, & 0 < y<1 , \theta> -1 \\ 0, & ...
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1answer
47 views

What is a “recurrent model” in forecasting

In this book, there is a chapter titled Recurrent Models (you can see that chapter in Google books) but it's very short and some parts are not very clear to me. Recurrent Models seem to refer to a ...
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1answer
65 views

Weights of airplane passengers

Due to rise in average Americans' weight guidelines are provided for airlines, expecting that plane passengers in the coming season will have an average weight of 190 pounds (luggage and clothes, etc, ...
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1answer
39 views

Alignment algorithm

So I'm trying to figure out to calculate some sort of alignment of strength score for a group of people's selections on various values. In this case, there are 36 values, each person selects gives 9 ...
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2answers
772 views

Symbols and naming in confidence interval

Suppose to have a confidence interval for the mean on a large sample, i.e. $$\overline{X}-z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}} \le \mu \le ...
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331 views

Finding the pdf of $Z=XY$, where the joint pdf of $(X,Y)$ is known

I've been trying to figure this out for a while. Suppose $X$ and $Y$ are random variables with joint pdf (probability density function) $$ f(x,y) = \frac{x+y}{15}, \quad\text{for }x = 0,1,2,\;\text{ ...
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2k views

Finding the MVUE using Rao-Blackwell Theorem

The number of breakdowns Y per day for a certain machine is a Poisson random variable with mean $\lambda$. The daily cost of repairing these break downs is given by $C=3Y^2$ If $Y_1, Y_2, ..., Y_n$ ...
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1answer
83 views

Binomial distribution probability

A married couple decided to have $5$ children. Based on gene history, probability that any one of their children will need to wear eye glasses, independent of sex, is $60$%; probability that a child ...
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1answer
42 views

Bootstrap-related issue

Say I re-sample $N$ items with replacement from a numbered item sample of size $N$. What is the average number of data items that are not selected in each such sampling?
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68 views

AP statistics question

A bag contains 2 black and 2 white marbles, and there is a supply of extra marbles of each color. A move consists of randomly drawing 2 marbles from the bag. If the marbles are the same color, they ...
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1answer
381 views

How would you approach this problem on the Bayes theorem?

I've been reading a book on Statistics and I could COMPLETELY understand all of its text. It basically explained the bayes theorem and what priors were, what posteriors were etc. But then in the ...
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1answer
47 views

Clarification about Stochastic distribution

What does this mean - "the x distribution is stochastically smaller than the y distribution"?
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216 views

Formula for confidence interval in multi-variable regression

What is the formula for calculating the confidence interval for the expected value of $\hat{y}$ in a multi-variable regression model. In other words, I'm looking for the following formula just for ...
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1answer
309 views

Binomial Distribution Parameter & Probability

A married couple decided to have $5$ children. Based on gene history, probability that any one of their children will need to wear eye glasses, independent of sex, is $60$%; probability that a child ...
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1answer
205 views

Showing a moving average is strictly stationary if underlying sequence is strictly stationary.

Just as the title suggests, this is my problem: Let $Z_t$ be a strictly stationary sequence. Define $X_t = Z_t + \theta Z_{t-1}$. Show that this sequence is also strictly stationary. Here's my ...
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1answer
450 views

Normalizing a normally distributed vector to unit length

If I have a random vector $\mathbf{y}$ generated from multivariate gaussian distribution $\mathcal{N}(\mathbf{0}, \mathbf{C})$, then I normalized it to unit length, which is, $$\mathbf{y} \sim ...
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Given a data set, how do you do a sinusoidal regression on paper? What are the equations, algorithms?

Most regressions are easy. Trivial once you know how to do it. Most of them involve substitutions which transform the data into a linear regression. But I have yet to figure out how to do a ...
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0answers
139 views

More efficient estimator?

I have this kind of problem: The market share $Z$ of a company is estimated in two independent survey polls. The sample sizes are $n_1=500$ and $n_2=2000$ with corresponding observed shares $p_1$ and ...
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2answers
128 views

Is it possible to find a 2D distribution function such that the higher order moments always exist?

Is it possible to generate a 2D distribution function $f(x,y)$ with function supports specified as $[-a,a]$ and $[-b,b]$ for $x$ and $y$ respectively, such that it always has moments which are NON ...
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1answer
543 views

Show these two equations for variance are equivalent

This is probably not too difficult, but I am having a lot of trouble with it. I am trying to show the following is true: $${\sum (x_i - \bar x)\over n-1} = {\sum x_i^2 - {(\sum x_i)^2\over n}\over n ...
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1answer
119 views

Equiprobable model for Pearson's goodness-of-fit method.

There is a very long introduction to this problem. I can provide this if needed but for now I will stick with the actual question. "A question of interest to the researchers was whether there were ...
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1answer
321 views

Calculate Statistics (Check if the answers are correct)

Calculate the statistics below using the following data on a sample of the variable $X$: Data ($X$ sample) = $\{9, -1, 7, 0, -2, 5, 4, 9, 5 \}$ Using the sample data, calculate: Mean; Median; ...
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1answer
40 views

How to calculate this permutation? [duplicate]

How many ways can UNC, Duke and Florida State finish 1-2-3 in the AAC regular season rankings? Please show work. Thank you.
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108 views

Is this a Combination or a Permutation?

Help Calculate: A car dealer has 3 body styles, 8 exterior colors and 2 interior color schemes. How many different cars are there? Please show work. Thank you.
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55 views

Is my answer to this combinatorics question correct? Counting the number of functions where $F(a) = F(b)$?

Let $A = \{a, b, c, d, e, f\}$ and $B = \{1, 2, 3, 4, 5\}$. How many functions $F$ from $A$ to $B$ are there such that $F(a) = F(b)$. I looked at it like this: If $F(a)$ and $F(b)$ are equal, that ...
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1answer
181 views

Completeness, Sufficiency and MLE of size n random samples of a joint distribution

Let $(X_1, Y_1), (X_2, Y_2), \dots , (X_n, Y_n)$ be a random sample of size $n$ from the continuous distribution with joint pdf $f_{X, Y}(x, y|\theta) = \frac{1}{\theta y}e^{-\frac{x}{\theta ...
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2answers
211 views

Using sufficiency to prove and disprove completeness of a distribution

Let $X_1, \dots ,X_n$ be a random sample of size $n$ from the continuous distribution with pdf $f_X(x\mid\theta) = \dfrac{2\theta^2}{x^3} I(x)_{(\theta;\infty)}$ where $\theta \in \Theta = (0, ...
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1answer
272 views

Bayes Estimator

Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf: $f_{X}(x|\alpha,\beta) = ...
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1answer
236 views

Sufficient Statistics and Maximum Likelihood

Let $(X_1, Y_1),\ldots, (X_n, Y_n)$ be a random sample of size $n$ from the continuous distribution with joint pdf: $$f_{X, Y} (x, y\mid\theta) = \frac{1}{\theta y}\exp\left(\frac{-x}{\theta ...
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1answer
325 views

what's the distribution of the inverse of a random variable that follows a negative binomial distribution?

I was studying the method of moments estimation of parameters, and I encountered the following problem. I have a geometric distribution as following: $P(X=k) = p(1-p)^{k-1}$, and a sample size of n, ...
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1answer
1k views

Discerning The Set Of Values For A Random Variable

The question is: For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. a. $X=$the number of unbroken eggs ...
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295 views

Variance of a function of a normal random variable

I want to define a new random variable $f$ as a function of a normal random variable $v$: $$f(v)=\begin{cases}C&\text{if } v\ge C\\ \gamma v &\text{otherwise}\end{cases}$$ where $v\sim ...
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1answer
507 views

Finding an efficient estimator for $ \beta $ in a sample of $ n $ random variables having the $ \text{Gamma}(\alpha,\beta) $-distribution.

Problem: Suppose that we have i.i.d. random variables $ X_{1},\dots,X_{n} \sim \text{Gamma}(\alpha,\beta) $, where $ \alpha > 0 $ is known. Find an efficient estimator for $ \beta $. Recall ...
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1answer
123 views

Probability of A given B & C?

I would like to sum the probability that given 3 (potentially biased) die rolls, all 3 rolls will be different-- what is the correct way to do this? So far, I have: $$ 1 - ...
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1answer
308 views

5 cards / Joint Probability Function

From a deck of playing cards, you take out 5. The random variables X and Y denote the number of "aces" and "queens" in the sample, respectively. Find the joint probability function of X and Y, and ...
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1answer
91 views

Is this a Permutation or a Combination?

To win a lottery, you must pick the winning 3 numbers from the integers 1-9 (no repeat numbers). What is the probability of winning the lottery by choosing the correct 3 numbers? I think its a ...
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1answer
36 views

Combination problem?

How many way can UNC, Duke, and Florida State finish 1-2-3 in the AAC regular season rankings? Would I have to find out the AAC'S regular season rankings?