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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Calculating rate of time consumption based on NBA player impact estimate

I'm building a game. I'm looking for help calculating the rate of consumption based on a player's impact estimate (PIE). PIE measures a player's overall statistical contribution against the total ...
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15 views

Coin Toss Normal Distribution

In 200 tosses of a coin, 115 heads and 85 tails were observed. In this problem we will compare using the normal, the normal with the continuity correction, and then using the binomial. For (a) -- (c),...
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19 views

MLE of heteroscedastic model?

Given the regression model where our and are identically and independently distributed. I'm trying to find the MLE B-hat and the unbiased estimator sigma-hat^2. I haven't dealt with any models in ...
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1answer
13 views

Expectation of absolute sum of squared normal distributions

Let $u_i$ be a standard normal distribution for all $i$. All $u_i$'s are independent of each other. I want to compute the expectation of: $$| \sum_i u_i^2 \lambda_i |$$ Where $\lambda_i$ is real ...
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19 views

l1 regularization mathematical explanation [on hold]

I somewhat understand what l1 regularization is, however, the mathematical formula and how to use it are confusing me. I'm not really sure what a regularization term is and how I could apply it to a ...
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2answers
34 views

Why does a null-hypothesis have to have a definite value?

In hypothesis testing, why does the null hypothesis (H_0) have to have one defined value?
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30 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
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2answers
56 views

Variance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
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3answers
35 views

Can I use mean and standard deviation to spot outliers?

I have a list of measured numbers (e. g. lengths of products). Of these I can easily compute the mean and the standard deviation. Now, when a new measured number arrives, I'd like to tell the ...
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20 views

Statistical calculation of value of coins in a box

I woke up from a dream today that made me consider the following scenario: A grocery store has an electronic donation box. Good Samaritans slide coins into the donation box, and the donation box ...
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11 views

Trouble with Bayesian Hypothesis Test Equation

A passage from Wasserman's All of Statistics: The Bayesian approach to testing involves putting a prior on $H_0$ and on the paramater $\theta$ and then computing $\mathbb{P}(H_0 \mid X^n)$. ...
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1answer
31 views

Showing sum of squared residuals is zero?

I have the model $$y_i = B_0+\sum\limits_{i=0}^pB_kX_{ik} + e_i$$ I'm looking to show the sum of squared residuals is zero if $p = (n-1)$. I have tried expanding it quite in depth and I haven't been ...
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2answers
23 views

Estimating a random variable from repeated trials

I have an $n$ sided die and suspect that it is biased. I'm interested in the probability of rolling a $1$, so I roll the die $m$ times and count up the number of times I roll $1$, then divide the ...
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0answers
17 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
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1answer
38 views

Showing Residual Sum of Squares for Multiple Linear Regression is 0

Problem: I have the linear regression model: $y_i=\beta_0+\sum_{k=1}^p \beta_kx_{ik}+\epsilon_i$ where $\epsilon_i\sim N(0,\sigma^2)$, for $i = 1,2,\ldots ,n$. I want to prove that the residual sum ...
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17 views

How to best compare two different time series with different frequencies

Lets say I have two time series $X_t$ and $Y_{t,q}$. As an examples, lets say $X_t$ is a series that measures year over year changes in the level of output of a good (say number of widgets). So $X_t = ...
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0answers
16 views

Unusual graph measure

Integrated information theory of consciousness is a complex mathematical model of information transfer in neural networks. Some of its conclusions are obvious: neither fully disconnected nor the ...
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12 views

Posteriori probability

I have two categories: A and B There are many experiments. In each experiment, there are two options: Result=$r$ will be added to A. $r$ $\in \mathbb{Z+}$ $r$ will be added to B. $r$ $\in \...
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14 views

Proof that SSR = SST when they have the same degree of freedom? [on hold]

Intuitively, this makes sense, but is there any way to prove it?
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23 views

Estimator bias and consistency

Let $x_1, x_2, \ldots,x_n$ be a simple random sample from a random variable $X$ with support $\{0,1,2,3,4\}$ and probability function $p(0)=\frac{5}{12}(1-\lambda)^2$, $p(1)=\lambda$, $p(2)=\lambda(1-\...
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1answer
26 views

Why does MLE work for continuous distributions?

In the attachment below you can see the definition of the likelihood function. Likelihood 1) Whilst the explanation of why the whole max likelihood method is viable for discrete distributions is ...
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19 views

standard deviation above or below the mean [on hold]

The following is data of 10 patients in a hospital 165,135,151,155,158,146,149,124,162,173 If a patient is selected randomly, find the probability his/her data exceeds one standard deviation above ...
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35 views

definition of a chi-square-distributed random variable in terms of independent N(0, 1)-variables [on hold]

What is the definition of a $\chi^2$-distributed random variable in terms of independent $\mathcal N(0, 1)$-variables? What is the relationship of a $t$-distribution with a standard normal and $\chi^2$...
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13 views

A question about a Jacobian in a proof of Neyman's factorization theorem

everyone. Would you be so kind and explain me the role of a Jacobian in the proof below (Picture attached, source wikipedia: https://en.wikipedia.org/wiki/Sufficient_statistic)? My knowledge is that ...
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1answer
32 views

Interquartile range to find out outlier & get perfect Standard deviation

I have one population dataset - 200, 330, 675, 999, 1200, 3000, 25000 For this dataset IQR = 3000 - 330 = 2670 Also we will ...
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24 views

Find the autocorrelation of y[n]=x[2n] in terms of the autocorrelation of x

Given that the autocorrelation of x is: $R_{xx} = $sin($\frac {\pi}{2}n)/(n\pi)$ I've tried to find the autocorrelation sequence but got confused about how to deal with the extra factor of 2 in the ...
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2answers
21 views

basic standard deviation question

The weights of newborn children in the United States vary according to the normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The government classifies a newborn as having ...
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1answer
7 views

Why is the size of a composite hypothesis defined as a supremum and not an infimum?

According to wikipedia: In the case of a composite hypothesis, the size is the supremum over all data generating processes that satisfiy the null hypotheses. $$\alpha = \sup_{h\in H_0} P(\...
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1answer
18 views

Finding the method of moments estimator for the Uniform Distribution

Let $X_1, \ldots, X_n \sim \text{Uniform}(a,b)$ where $a$ and $b$ are unknown paramaters and $a < b$. (a) Find the method of moments estimators for $a$ and $b$. (b) Find the MLE $\hat{a}$ ...
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0answers
25 views

Find an estimator by using the method of moment

Let $X$ be a discrete random variable with density function: $$p(x;\theta)=\left(\frac{\theta}{2}\right)^{\lvert x\rvert}(1-\theta)^{1-\lvert x\rvert}$$ where $x\in\{-1,0,1\}$ and $\theta \in[0,1]...
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22 views

Transformation of confidence interval

Let $X\sim \mathcal N(\mu,\sigma^2)$. Given a simple random sample I know that $$\left[\bar X - z_{1- \alpha/2} \frac{\sigma}{\sqrt{n}},\bar X + z_{1- \alpha/2} \frac{\sigma}{\sqrt{n}}\right]$$ ...
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2answers
102 views

How to maximize Std Dev given a range of possible values, a number of values, and a specific mean?

(I'm asking here and not stats.stackexchange because I'd like a mathematical proof of this) In this question: Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values;...
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1answer
39 views

Real analysis reference for statistician

I'm a undergraduate statistics student, I think that learn Real Analysis can be useful to me in some points, can anyone suggest a introductory book for self-study ? I'm already multivariate calculus, ...
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25 views

how many samples needed to obtain an estimate with a given confidence interval

Suppose an urn contains N balls of different colors. I do not know the colors nor the distributions, and I wish to determine the fraction of red balls in the urn, (R/N), to within p% with C confidence....
3
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1answer
55 views

Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
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0answers
15 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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1answer
19 views

Expected value conditioned [on hold]

Given $X_1, \ldots, X_n$ r.s.s. from a random variable with probability function $$f_{\theta}(x)=\frac{1}{\theta}\text{ for }x=1, \ldots, \theta$$ Let $T_1=2X_1-1$ and $T_2=X_{(n)}$ (maximum of $X_1, \...
2
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1answer
19 views

Distribution of the minimum

I have the following problem, given a random variable $X$ with density $$f(x)=2x\text{ for }x\in(0,1)$$ and a r.s.s. $X_1, X_2, X_3$. I have to calculate the probability that $X_{(1)}=\min\{X_1,X_2,...
2
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1answer
16 views

Deriving the mean of the Gumbel Distribution

I'm trying to determine an expected value of a random variable related to the Gumbel/Extreme Value Type 1 distribution. I think the answer follows the same process as expected value of the Gumbel ...
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1answer
18 views

Find efficient estimator

Let $x_1$, $x_2$,...,$x_n$ be a simple random sample from $X\sim \mathcal N(\mu,\sigma^2)$. Which estimator is more efficient between $T_1=\frac{1}{n}\sum_{i=1}^n X_i=\bar X$ and $T_2=2\bar X-X_1$? ...
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0answers
22 views

Quotient of two Gaussian densities

The matrix cookbook contains formulas for the product of two multivariate Gaussians, but doesn't appear to contain formulas for the quotient of two Gaussians. $$ \frac{\mathcal{N}(\mathbf{m}_1, \...
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1answer
33 views

What does the 'probability of incorrectly rejecting Ho' mean?

In the following A-Level S2 question, the second part asks you to state the probability of incorrectly rejecting Ho using this critical region I found the critical region: X ≤ 3 ∪ X ≥ 16. And ...
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1answer
18 views

Basic question about how to calculate a variance

How to calculate the variance of $X$: $X= 1 w.p. 0.5; -1 w.p. 0.5 $ I was trying $Var(X)=E(X^2)-(EX)^2=E(X^2)-0=X^2P(X^2=-1,-1)+X^2P(X^2=1,1)+X^2P(X^2=-1,1)+X^2P(X^2=1,-1)=-1/4-1/4+1/4+1/4=0$ ...
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1answer
30 views

Given the correlation matrix, estimate the value of a random variable based on the value of other random variables. [on hold]

A process generates $N$ random variables $(X_i \mid 1 \leq i \leq N)$. The process is run $K$ times, and the values of each random variable $X_i$ is observed. Based on this data, the following ...
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27 views

Prediction for time series

Suppose we have several weeks (say, 30) in a row and number of advertisements sent on e-mails on each week and number of the following orders of the product. How can I estimate the necessary number ...
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12 views

ACF of aggregated AR(1) processes

Consider the sum of AR(1) processes \begin{equation} \sum_{i=1}^N x_{i}(t) \end{equation} where $x_i(t) = \alpha_i x_i(t-1) + \epsilon(t)$ and the coefficients are sampled from a Beta distribution, \...
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2answers
25 views

How to normalize and inverse a vector so it sums to 1 ?

I understand how normalization works. You sum up the individual values of the vector, you divide each value by the sum, and voila... they sum to 1. Why doesn't it work when you subtract them from ...
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15 views

Name of the field of study that details extrapolation of a series based on subset sample data

Apologies if my notation and/or terminology is way off - I'm not well versed in mathematics. I'm looking for the name of the field of mathematics that might help me solve my problem. Here's my problem:...
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20 views

How to use the log-sum inequality to prove convexity of KL-divergence?

I'm trying to read up on information theory, and found the following: http://homes.cs.washington.edu/~anuprao/pubs/CSE533Autumn2010/lecture3.pdf Which states that the convexity of KL-divergence can ...
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1answer
30 views

Logistic regression giving negative predictions?

I am doing a binomial-family logistic regression, but my predictions on the datasets are almost all negative. Is this because the resulting predictions are actually log-likelihoods, not pure ...