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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18 views

showing that a matrix has repetitive values?

Here my primary aim is to calculate the stationary distribution of a DTMC using left-eigen values i.e, $ \pi = \pi*P$. But for some matrices, I observe that some states a same stationary probability. ...
1
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0answers
19 views

Testing the population mean

Given a population size and a sample, how can i test if the population mean is above a certain value? Similarly, can i find out what what value the sample mean has to be to say that the population ...
2
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1answer
567 views

Weighted Standard Deviation for Histogram Bin Height

I'm plotting some binned data in the form of a histogram. Say I have 10 data points, each composed of a bin to be placed in, and then a "height". Then I might have something like: Bin Height 0 - ...
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1answer
26 views

Calculating approximate growth from three numbers [closed]

I have a set of three numbers 3600, 5200,12000; how do I calculate an approximate 4th number ...
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0answers
15 views

Correct statistic test(s) for resting energy expenditure (REE) vs estimated weightloss percentage (EWL)

Hello I am a student wanting to do a statistics test to compare the following: REE (initial) vs %EWL after one year. REE (initial) vs weight loss after one year. Change in REE (initial to one year) ...
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1answer
18 views

How to create a fair dynamic scoring system? [closed]

I am currently in the process of creating a game consisting of a fixed set of tasks of varying difficulty. Each player gets the same set of tasks to choose from and is awarded a certain number of ...
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0answers
22 views
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1answer
10 views

Confidence interval for a density function parameter

I am trying to solve this problem: Given $X_1, \ldots, X_n$ a random sample of a population of random variables with p.d.f. $f(x, \theta) = e^{-(x-\theta)} I(x)_{x\geq\theta}$, find a confidence ...
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1answer
58 views

Probability of getting n heads when n is very large [closed]

How to show that the probability of getting n heads when 2n times coin is toss is very small? Moreover, how to show that the probability of more than n heads is close to 0.5?
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1answer
35 views

Use of language on wikipedia - what kind of distribution?

I have an interesting problem and was wondering whether anyone would be able to point me in the right direction. I am wondering whether the use of a word in the english language on Wikipedia is ...
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1answer
17 views

Estimating a parameter using the maximum likelihood-method and the method of moments

Let $X$ be a random variable that has a density function of the form $f_X(x) = (p + 1) x^p 1_{[0, 1]}(x), x \in \mathbb{R}$ where $p > 0$ is an unknown parameter. I now want to make an educated "...
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0answers
27 views

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

Find the autocorrelation of $y[n]=x[2n]$ in terms of the autocorrelation of $x$, given that the autocorrelation of $x$ is: $$R_{xx} = \frac 1{n\pi}\sin\left(\frac {\pi}{2}n\right).$$ I've tried to ...
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23 views

poisson process question.

The following calculations arise in certain highly simplified models of learning processes. Let $X_1(t)$ and $X_2(t)$ be independent poisson processes having parameters $\lambda_1$ and $\lambda_2$, ...
3
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1answer
53 views

l1 regularization mathematical explanation

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 ...
2
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2answers
394 views

Algebraic manipulation of normal, $\chi^2$ and Gamma probability distributions

If $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$, then $$ \frac{n - 1}{\sigma^2}S^2 \sim \chi^2_{n - 1} $$ where $S^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i^2- \bar{x})^2$, and there's a direct relationship ...
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0answers
32 views

What is the computational complexity class of thermal photon statistics

I would like to know the computational complexity of the following formula for the variance of thermal photon statistics. $$P(n)=\sum_{d=1}^D\prod_{m=1}^M\frac{1}{(1+\langle n_m \rangle)(1+\langle ...
3
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1answer
50 views

Should principles be proved? [closed]

I was wondering if one needed to prove principles. E.g., likelihood or condionality principles in Stats. Thank you!
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0answers
28 views

A “sloppy” proof of Neyman's factorization theorem

Could you please explain why the attached proof is called "sloppy"? What is wrong with it? Thank you!
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2answers
39 views

Find the Moment Generating Function $Y$. What is the distribution of $Y$?

Let $X_1$ and $X_2$ be independent normal variables with means 2 and 5 and variances 9 and 1. Let $Y = 3X_1 + 6X_2 - 8$. Find MGF. What is the distribution of $Y$. attempt: Im not sure about ...
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1answer
70 views

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? I have got the total no of events by using 64 C 2. But I am unable to find the numerator(no. ...
3
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2answers
28 views

intuition behind having a unique regression line

I understand this mathematically. we have function of 2 variables represents the sum of square errors. We have to find the $a$ and $b$ that minimize the function. there is only one minimum point. But ...
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1answer
2k views

Statistics - Z-score confusion.

For z score, you are taking the sample value subtracting population mean and dividing it by std deviation. Is that correct so far? Now, the "sample value" is defined by an equation. In my scenario, I ...
2
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0answers
64 views

Asymptotic behaviour of sums of covariances of RVs with LRD

Our assumptions are: $\left(X_t\right)_{t\geqslant 0}$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\...
4
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1answer
505 views

Determine whether ARMA(p,q) is stationary and/or invertible?

Determine whether an ARMA(p,q) process is stationary and invertible such that $y_t = \sum_{i=1}^{p} \phi_i y_{t-i} + \sum_{i=1}^{p} \theta_{i} \epsilon_{t-i}$ with the restriction that $\theta_{0} = ...
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4answers
83 views

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue? I am helpless regarding this. I don't know how to solve it....
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1answer
74 views

Show that Cov(T,T')=Var(T) [closed]

Let $X_1,...,X_n$ be a sample, all with mean $\mu$ and $Var(X_i)<\infty$. Let $T(X_1,...,X_n)=\sum_{i=1}^na_iX_i$. If T is the UMVUE of $\mu$ and T' is another linear unbiased estimate of $\mu$, ...
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0answers
24 views

What is the relationship between the function $\mathbb{E}(Y \mid X = x)$ and linear regression?

Consider the function $$ r(x) = \mathbb{E}(Y \mid X = x) $$ This has been called the regression function in a textbook I'm using. I'm trying to figure out the relationship between this function ...
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2answers
33 views

How to generate a random variable $r_i$ such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$

How can I generate $r_i$ for $1 \leq i \leq n$, such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$, where $\sigma_i^2$ is the variance of $r_i$ and, $\chi^2_{n,\alpha}$ is a chi-...
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2answers
37 views

Probability question with a radio competition

I'm quite new to statistics and I'm going through a few exam questions but I am a bit stuck on this one: ...
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1answer
38 views

Relationship between averages of $x^tx$ and $xx^t$ for column vector $x$

If we have data set $x$ as $m$ of $n \times 1$ vectors, and we know the average over index $m$ of $xx^t$ is $<xx^t> = C$, where $C$ is $ n \times n$ matrix. What is the average of scalar $ x^...
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1answer
29 views

Uniformly Choosing a number from a range [closed]

May you please help me how I can choose uniformly a number from a range. I have to use this for trust evaluation in social networks such as the following clause: Each user has a quality ...
3
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1answer
628 views

Pearson's Chi Squared / Cochran–Mantel–Haenszel test analog to N-way ANOVA

A test was given to two sets of students, CONTROL and EXPERIMENT, that had question A and question B. I want to know if students who got question A right were more likely to get question B right, and ...
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0answers
35 views

Common Estimates Suggestions

Consider two Markov Chains $X-Y_1-Z_1$ and $X-Y_2-Z_2$ defined on same alphabet space $\mathcal{X}$, such that $Z_1= g_1(Y_1)$ and $Z_2=g_2(Y_2)$ for some functions $g_1,g_2$. Assume further that ...
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34 views

Lottery machine probability

Assume we have a lottery machine where you press a button and it returns one of 5 motifs in one of 5 colours. Each of these also has a chance to be gilded. Assume I have a dataset containing outcomes ...
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2answers
40 views

Derive the value of this probability analytically

Forgive me if this question is very basic but I genuinely tried to search around including this site and could not find anything that I could adapt to my understanding. ...
1
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1answer
1k views

Comparing annualised volatility from monthly and annual data

I fear there is a very simple answer to this question and its killing me that I can't see it. I am interested in calculating historical volatility: I have monthly index values starting in Jan 2005 ...
2
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1answer
17 views

Find the PMF for number of heads following the first tail on a four consecutive coin toss expriment

Suppose a fair coin is toss four times consecutively. Find the PMF for random variable of number of heads following the first tail. My take: Let random variable $X$ be the number of heads in this ...
0
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1answer
23 views

The relation of correlation coefficient of the sum of two vectors.

Does the correlation coefficient of the sum of two vectors between the correlation coefficient of each of them. Suppose I have three vectors $x_1,x_2,x_3$. The correlation coefficient of $x_1$ and $...
0
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1answer
38 views

What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
1
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1answer
40 views

If I know the mean, median, mode, and so on how do I determine the standard PDF that most fits

I have been playing a game and it drops currency with a drop rate that has an expected value of $1/750$ and nothing else. I was able to keep track of 3000 drops and times between them. I can then find ...
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1answer
31 views

Simple Question on interpretation of Poisson dist. as approximation to Binomial dist.

I have a question regarding to the notion of using Poisson dist as an approximation to Binomial dist. I can easily prove $\lim_{n\rightarrow\infty} P(X=k) = P(Y=k)$, where $X \sim$ Binomial(n, p) and $...
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0answers
9 views

What should be expected values of weights in statistics (specifically a set of biweight weightings)

I am working on a project that involves taking the biweight sample variance of a velocity dataset. This is defined as $\sigma_{BI}^2 = N\dfrac{\sum_{|u_i|<1}(1-u_i^2)^4(v_i-\bar{v})}{D(D-1)}$ ...
3
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1answer
45 views

Variance of sum of linear combination

I want to calculate the variance of a sum of linear combinations, so $$\operatorname{Var}\left(w'R_1 + w'R_2\right)$$ where $w$ is a $N\times 1$ vector and both $R_1$ and $R_2$ are $N\times 1$ ...
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2answers
60 views

Why does the normal distribution describe data collected in real life so well? [closed]

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$ Is there any intuition behind choosing $e^{-x^2}$ instead of some other function?
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1answer
723 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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18 views

Problem in finding introductory material (matrix spectra)

I am looking for introductory material on: 1) matrix eigenvalue spectra and useful matrix algebra theorems that can be applied in the field. 2) Statistics of random matrices (i.e. ensembles, ...
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2answers
37 views

Probability of incorrectly spelling a word

I'm currently trying to teach myself Statistics and have an exam question that I need a bit of help on: ...
2
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0answers
13 views

Derive the Hat Matrix to map actual response to estimated resposne

In order to measure the quality of a regression we can calculate the Hat Matrix. Using it we can estimate the response variable as if we used the predictor variables to regress them. For linear ...
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1answer
14 views

loss function similar to normal density

let $$L_\epsilon(x,p) = -\frac{1}{\sqrt{\epsilon}}\exp\left\{-\frac{(x-p)^2}{\epsilon}\right\}$$ be a loss function. given a random variable $X$ with density $f$ (possibly restricted), the risk ...
4
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2answers
130 views

What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and $\...