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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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
20 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
39 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 = ...
2
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1answer
22 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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13 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? [closed]

Intuitively, this makes sense, but is there any way to prove it?
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24 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
27 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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20 views

standard deviation above or below the mean [closed]

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

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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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
33 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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0answers
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
22 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 $\hat{a}$ ...
2
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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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23 views

Transformation of confidence interval

Let $X\sim \mathcal N(\mu,\sigma^2)$. Given a simple random sample I know that $$\left[\bar X - z_{1- \alpha/2} \frac{\sigma}{\sqrt{n}},\bar X + z_{1- \alpha/2} \frac{\sigma}{\sqrt{n}}\right]$$ ...
3
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2answers
108 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....
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1answer
56 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
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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 [closed]

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, \...
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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. [closed]

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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0answers
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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0answers
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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0answers
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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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21 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 ...
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1answer
60 views

If a student who received an A in probability is chosen at random, what is the probability that he/she also received an A in calculus?

This question has been asked before but the solution given was incorrect.(see here) A prerequisite for students to take a probability class is to pass calculus. A study of correlation of grades for ...
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1answer
22 views

Probability of an event not occurring given it frequency of occurence

Given: There is an event that can occur 0 or more times during a given discrete process. An observation that the average number of events per process is p. Questions: Using only p above is it ...
1
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1answer
42 views

Can I ignore multicolinearity problem if all the regression coefficients are highly significant?

Can I ignore multicolinearity problem if all the regression coefficients are highly significant? My data is large enough and all the resulting coefficients are significant enough in less than 0.01 ...
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1answer
42 views

Complicated probability question [closed]

There are 20 empty present boxes numbered from 1 to 20 are placed on a shelf, there are 4 men standing in front of the shelf. Each one asked to pick in his mind 3 numbers without telling any one then ...
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0answers
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prove that Gaussian kernel function is a kernel function

How to show that the Gaussian kernel : $K(x,y)=\exp(0.5||x-y||^2)$ is a kernel function? (meaning it is a Gram matrix of some transformation of $x$ and $y$) I know that to prove that a function is a ...
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2answers
45 views

Sum over Binomial mass function

In Casella and Berger Book (Statistical Inference), exercise 2.40 is $$\sum_{k=0}^x {n\choose k}p^k(1-p)^{n-k}=(n-x){n\choose x}\int_0^{1-p}t^{n-x-1}(1-t)^xdt.$$ If I replace $x$ by $n$ then LHS ...
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1answer
39 views

Why is the CLT stated like it is?

The CLT says that given finite variance of iid RVs, we have $$\sqrt{n}( \bar{X} - \mu) \rightarrow \mathcal{N}(0,\sigma^2),$$ but if this is true, then $\bar{X} - \mu$ should converge to $\mathcal{N}(...
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1answer
26 views

Consistent Estimator and Convergence Variance

I was practicing an exam, and came along this question: A consistent estimator converges in probability to the true parameter value. Therefore, the variance of such an estimator converges to zero ...
2
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1answer
31 views

Proof by contradidction that the mean of a set cannot be greater than the greatest value in that set.

I want to prove that given a set of values $x_1, x_2, ..., x_n$, the mean of those values cannot be greater than the greatest of those values. Let the mean $\frac{x_1 + x_2 +... + x_n}{n} = a$ ...
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1answer
25 views

Notation for minimum

I'm reading Huber's Robust Statistics right now, and at the beginning of Chapter 3, he writes the following notation: $\sum \rho(x_i;T_n) = \min!$ Similarly, a few lines down, he writes: $\sum \rho(...
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3answers
76 views

chances of a group being all of the same sex [closed]

I was wondering, if there are 10 girls and 10 boys in a classroom, and they were randomly assigned in groups of four, what are the chances of there being a group with all people inside it the same sex ...
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1answer
17 views

Find Standard Deviation given Population and Proportion Info [closed]

I know the answer is D but how? There is a 30% chance of missing a flight due to long lines at security. If there are 120 people waiting in the security line to make their flight, what is the mean ...