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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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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11 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
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18 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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18 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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2answers
55 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 ...
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20 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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7k views

How do you calculate percentile?

Say I have the following numbers: 10 10 9 5 4 4 4 1 How do i go about calculating the percentile for each score? Is there a standard formula for figuring this ...
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1answer
374 views

Multiple regression degrees of freedom f test.

I'm finding conflicting information from college textbooks on calculating the degrees of freedom for a a global F test on a multiple regression. To be absolutely clear, assume there are 50 ...
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30 views

Simple question about showing independence

How does one show that it is possible for a random variable $Z$ to be independent of $A$ but also not independent of $X$ where $X=1\{A>B\}?$ Under what circumstances can this be true, ignoring the ...
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19 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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544 views

median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
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32 views

Finding UMVUE for Poisson distribution using Rao Blackwell

Let ${X_1,X_2,... ,X_n}$ be a random sample from a Poisson distribution with parameter $\lambda$.Let $\gamma(\lambda)=P(X<=1)$. Find UMVUE for $\gamma(\lambda)$. This is my attempt: First I ...
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3answers
1k views

What situation calls for dividing the standard deviation by $\sqrt n$?

While doing my homework and checking my answers with the book's answers I noticed that sometimes the standard deviation is divided by $\sqrt n$ where $n$ is the sample size. I'm a little confused. For ...
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1answer
73 views

Statistics - Approximating Poisson Distribution

Y, the number of accidents per year at a given intersection, is assumed to have a Poisson distribution. Over the past few years, an average of 36 accidents per year have occurred at this intersection. ...
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1answer
26 views

How to calculate the HHG (Heller Heller Gorfine) correlation

HHG (A consistent multivariate test of association based on ranks of distances) is introduced in: Heller, R., Heller, Y., & Gorfine, M. (2012b). A consistent multivariate test of association ...
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2answers
3k views

Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
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8 views

What is the minimum number of data points required to use ARIMA(p,d,q)? [migrated]

I'm using the $\mathrm{ARIMA}$$(p,d,q)$ model to predict future time series data in R (see also R's ARIMA documentation). Q: What is the minimum the number of data points we must have in the time ...
2
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1answer
32 views

When can we write $f(v)dv=f(E)dE$?

In statistical thermodynamics we write $$f(v)\,dv = f(E)\,dE$$ where $v$ is velocity and $E= \frac12mv^2$ is energy and $f$ refers to the distribution function Can someone explain the logic ...
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1answer
28 views

How to get the standard deviation of a given histogram (image)

I have a doubt: How to get the standard deviation of a given histogram? I'm looking for it on the internet. But I got Nothing. For example the case of this image below Thanks in advance!
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2answers
86 views

Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.

I have a problem from a homework which I've been struggling with. I normally wouldn't post homework here, but I've spent several hours trying to understand the correct way to solve this , to no avail. ...
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5answers
11k views

How to intuitively understand eigenvalue and eigenvector?

I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman. Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to ...
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1answer
247 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
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Statistical methods of calculating weights for criterion describing quality.

In a research of quality of different sound recording techniques volunteers where asked to grade samples using multiple criteria. Those criteria describes different aspects that directly contribute to ...
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1answer
85 views

Generating random variates in Excel

I am very confused with a question I have found in relation to Excel. I am hoping someone can help me do this or at-least give me direction in which I can figure out how to do this. So far I don't ...
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1answer
17 views

Infimum of Gamma distribution

Let $X$ be a Gamma random variable with the CDF $F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete ...
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51 views

Determining whether random variables are independent

If I have two random variables as follows: 1) A Gaussian distribution of wifi signal strengths at a known point 2) A Gaussian distribution of wifi signal strengths at an unknown point (Note that ...
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1answer
28 views

Cohan's Kappa different class sizes

For my research I need to look at the Cohan's Kappa measure to find out if there is agreement. I asked four people to cluster 38 words and I got the following results: Person 1: 6 clusters Person 2: ...
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3answers
252 views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
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2answers
19 views

Find probability for distribution of several random variable

let $f(x,y)=k$ if $x >0, y>0, x+y <3$ and $0$ otherwise. Find $k$, find $P(x+y \leq 1), P(Y>X)$ I know that $0 <y<3-x$ but I also need to find the upper bound of $x$ in order to do ...
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1answer
252 views

Get a Fisher information matrix for linear model with the normal distribution for measurement error?

For given linear model $y = x \beta + \epsilon$, where $\beta$ is a $p$-dimentional column vector, and $\epsilon$ is a measurement error that follows a normal distribution, a FIM is a $p \times p$ ...
3
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1answer
50 views

Find the probability that you have three of a kind after you pick 5 cards from a regular 52-card deck.

How would I go about finding this? I was thinking that, since this is a permutation, you would have 52!/(52-5!)? But how would that account for 3 of a kind?
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1answer
343 views

How to use point estimation on an interval? Statistics question

I got part a and part b just fine, but I am confused on how to do part (c), (d), and (e). I don't really understand what the question is asking, nor do I understand how to do that sort of interval ...
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1answer
34 views

Covariance of random numbers not near zero

I'm learning about making an efficient bounding volume for a point cloud and this touches on statistics: an area of mathematics I'm very unfamiliar with. The book talks about how to compute ...
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2answers
28 views

Find probability in card question

Suppose that a test for extrasensory perception consists of naming (in any order) $3$ cards randomly drawn from a deck of $13$ cards. Find the probability that by chance alone, the person will ...
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1answer
23 views

Probability of poisson distribution

Let $X$ be the number of cars per minute passing a certain point on Sunday from 8AM to 10AM. Assume that $X$ has poisson distribution with mean $5$. Find the probability observing $4$ or fewer car in ...
3
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1answer
258 views

Covariance- v. correlation-matrix based PCA

In principal component analysis (PCA), one can choose either the covariance matrix or the correlation matrix to find the components. These give different results because, I suspect, the eigenvectors ...
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2answers
50 views

What is the expected value of max(x-y,0)?

For a normally distributed $X$ (with mean zero and variance $\sigma^2$) and a constant $Y$, I need to know $\mathbb{E}(\max(X-Y,0))$ I think that this should be $\mathbb{P}(X<Y)\times 0 + ...
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2answers
18 views

Estimating an unknown addition to data set

I have a data set with 11 points. The smallest point is 60, the largest is 122, and the average is 92.36. This is all that is known about the data set. I add one point (unknown size), such that there ...
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0answers
32 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
3
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0answers
26 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
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1answer
68 views

Independent Probability Events?

Events $A$ and $B$ in probability space $S$ are independent if the occurrence of one of them does not influence the occurrence of the other. In other words, $B$ is independent of $A$ if $P(B) = ...
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36 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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20 views

Distribution Significance

I am trying to determine if a distribution is significant given an overall population distribution. The population as a hole has a 4% distribution for my question. If a sub-group of my population has ...
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2answers
27 views

Find probability

Let $X$ (millimeters) be the thickness of the washer. Assume that $X$ has density $$f(x)= \begin{cases} kx & \text{if $0.9 \leq x \leq 1.1$};\\ 0 & ...
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0answers
16 views

Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
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8 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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37 views

Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...
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1answer
32 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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1answer
13 views

How to calculate possible permutations with varying constraints

I have created a random email generator. The email is put together with the following format: (random first name) (random separator) (random last name) (random integer) @ (random domain) 150 ...
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2answers
251 views

Mode from continuous distribution

I've continuous distribution set of values. For example: 0, 0.01, 0.012, 1.2, 5.33, 5.24, 5.38, 30.20, 30.21, 30.13, 30.12 I want to calculate most frequent ...