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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CLT for continuous functions of random variables

Let $(X_i)$ be a collection of zero mean, unit variance, real valued random variables (I do not assume that they are iid). Let $\mathcal H$ be a separable RKHS with a bounded kernel $k(x,y)$. Suppose ...
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15 views

Sample size given Type I and Type II errors

I have a normal distribution $X \sim N(\mu;0.002)$. We want to test if $\mu=10$. Determine the size of the sample if we want The probability of rejecting the hypothesis $\mu=10$ given that $\mu=10$...
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1answer
31 views

Why is this Markov chain's stationary distribution not (1/2, 0, 0, 1/2)?

I have the Markov chain 1, 0, 0, 0 1/2, 0, 1/2, 0 0, 1/2, 0, 1/2 0, 0, 0, 1 I understand how to build the system with which I am supposed to find that ...
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0answers
28 views

Exponential smoothing and variance

I have a time series, let's denote it as $X_t$ (a series of prices). We exponentially smooth the data to obtain another time series, an exponential moving average: $EMA_t$. My question is how do I ...
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1answer
15 views

Range of a random variable

A set of notes i'm working through gives the following definitions of the range of a random variable and then in addition a variation of the definition. Definition: The range of a random variable is ...
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1answer
24 views

Variance of Least Squares Estimator

Suppose a fit a line using the method of least squares to $n$ points, all the standard statistical assumptions hold, and I want to estimate that line at a new point, $x_0$. Denoting that value by $\...
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13 views

Using population mean and population std deviation to test range of values

Given: population mean μ = 156 and population standard deviation σ = 5.1. No more than 15% of values should be less than 150. What percent of values are actually less than 150? This is a homework ...
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7 views

Multiple surveys with different sample sizes for same population parameter

I don't have a very strong background in statistics but I have seen that results in experiments can be later on found to be statistically unsound. So I just want to make sure that my results are good, ...
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3answers
36 views

The probability of being paired with someone?

This question was presented to me last summer. It goes as follows: There are 8 people in a bus. All of them will be attending a camp, and will all be placed into rooms with one other person. There ...
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1answer
48 views

Why does the pmf for a Poisson Distribution Maximize at $x = \lambda$?

For a random variable $X$ s.t. $X$ has a Poisson distribution: $$ P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!} $$ The following graph seems to indicate that the maximum ...
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14 views

Bayesian equation: need for priors

As far as I understand, in the problem of Bayesian inference we have a random variable $y$ describing data, which is distributed according to some parameter $x$ via the known conditional distribution $...
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1answer
28 views

Chi Distribution Problems [closed]

I don't really understand how to solve these...could anyone help me out? 1)How to find the expected value of χ2(9) ? 2)We have a point A(x,y,z) in a 3 dimensional space. Each of the coordinates is ...
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4 views

PPK Target Values

I'm assessing specifications for a bunch of different products my company makes and determining whether or not the manufacturing processes are in control. To do this, we are calculating Ppk Values ...
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0answers
34 views

existence of some variance over chebyshev's inequality

From the basic knowledge I have, there must exist the variance for some epsilon greater than zero but less than 1 for which the Chebyshev's inequality holds. Now the scénario I have to verify using ...
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1answer
33 views

Proof of Chebyshev's inequality for a geometric random variable

I have learnt the Chebyshev's inequality for a continuous case like log-normal and normal distributions and in trying to understand the application I came across the question: For a geometric ...
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14 views

Find the critical value in Tukey's HSD

I'm trying to find the formula for finding the critical values for Tukey's HSD but I can't find any documentation on how to calculate the critical value based on the number of groups the type I error ...
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1answer
24 views

multiple linear regression model: scale the dependent variable y by a factor $c ∈ \mathbb{R}, c \neq 0$ [on hold]

In the multiple linear regression model $y = Xβ + u$, if you scale the dependent variable $y$ by a factor $c ∈ \mathbb{R}$, $c \neq 0$, how does the LS estimator $\hat{β}$ change? Does such a change ...
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31 views

Distribution on number of revisits in past $k$ steps of Markov chain

Consider a finite-state Markov chain with transition matrix $P$. The chain starts in a state chosen uniformly over all the states and runs indefinitely from there. We're going to examine only the $k ...
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Integrate Beta and Normal CDF mixture

Is it possible to integrate the following integral? $\int_0^1 y^{m-1}(1-y)^{n-1}\Phi\left(\Phi^{-1}(y)+\mu\right)dy$, where $m, n, \mu$ are constants and $\Phi(.)$ is the normal CDF Thank you
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1answer
16 views

Solving for a Conjugate Prior in search of MAP estimator

I am trying to prove that if a given random variable $X \sim Exp(\lambda)$ and $\lambda \sim Gamma(\alpha,\beta)$ hen $\lambda | X \sim Gamma(\alpha^{*},\beta^{*})$ for some parameters $\alpha^{*}$ ...
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3answers
68 views

Why do some pdfs y axis values greater than 1

I've been trying to wrap my head around different types of pdf. Do pdfs show relative probabilities? Take this example of the exponential: When lamba=1.5 p(0)=1.4 Obviously the probability of ...
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12 views

Equality of Quotients of Probabilities from Combinatorics

Sorry the title is so vague - I don't know how else to ask this. Essentially, what is a real-life example showing why: $\frac{_aC_k}{_nC_k}=\frac{_aP_k}{_nP_k}, where \ a<n$ is true?
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19 views

Kullback-Leibner divergence true distribution

I have an image with an object which I treat as 2-dimensional Gaussian random vector with mean equal to the center of the object surrounded by, roughly, 3-sigma ellipsoid. On the other hand I feed the ...
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1answer
34 views

On the sum of two independient normal random variables

Theorem. If $X$ and $Y$ are two independent normal random variables with means $a,b$ and variances $c,d$ respectly, the sum $X+Y$ is a normal random variable with parameters $a+b$ and $c+d$. My ...
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1answer
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Can I run a regression when both independent and dependent variables are all dichotomous?

I have conducted a survey where all my questions are asked in a dichotomous manner (Yes/No). Eg IV:"Are you a smoker?", "Are you obese", "Is your gender male/Female" etc. DV: "Have you ever had a ...
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1answer
14 views

How the value of denominator calculated here?

I found this example in a book and it has to find probability distribution as stated below: If a car agency sells 50% of its inventory of a certain foreign car equipped with side airbags, find a ...
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1answer
22 views

Calculating variance upon removing a value from the sample

We are given that a sample of $N=7$ has mean $\bar{X} = 5$, and a sample variance of $s^2 = 9$. One person, with a score of $11$ is removed from the data. What is the mean and variance for the new set ...
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17 views

hypothesis testing question.

Sorry for any English mistakes, Looking for a push in the right direction with this problem. ...... Intro: 2 researchers are testing a hypothesis separately.. $$H_0 : \mu = \mu_0, H_1 : \mu >\...
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14 views

$X$ and $Y$ are standardized r.v.s. Find $a,b,c,d$ such that $Z=aX+bY$ and $W=cX+dY$ are uncorrelated but still standardized.

Let $X$ and $Y$ be standardized r.v.s (i.e., marginally they each have mean $0$ and variance $1$) with correlation $\rho \in (−1, 1)$. Find $a, b, c, d$ (in terms of $\rho$) such that $Z = aX + bY$ ...
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23 views

GMM with full and diagonal covariances

I have Gaussian Mixture Model-- distribution with probability density function, that is a weighted sum of Gaussian probability density functions: \begin{equation} p(X)=\sum_{i=1}^k \omega_i\mathcal{N}...
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2answers
62 views

$n$ points be placed uniformly at random on the boundary of a circle of circumference $1$.. what's the $n$ arcs' length distribution?

Let $n$ points be placed uniformly at random on the boundary of a circle of circumference $1$. These n points divide the circle into $n$ arcs. Let $Z_i$ for $1 \le i \le n$ be the length of these ...
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10 views

Alternative to QDA and LDA on supervised classification

I have to do a supervised classification on a dataset made by 12 variables and 5 groups. The main problem is that my data are not gaussian so, i cannot make the quadratic and linear discriminant ...
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71 views

Esscher Transform extended

This problem is almost solved, dont get scared by the massive text The Esscher-transform is a well know tool in the financial section. I posted this in statistics also, since it relates to continuous ...
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1answer
47 views

Given sum of uniform random variables $Z_1 + Z_2 + \dots + Z_n =1$,what's the probability that $k$ R.Vs are at least $1/n$?

Given sum of uniform random variables on $[0,1]$, $Z_1 + Z_2 + \dots + Z_n = 1$, what is the probability that exactly k random variables are at least $\frac{1}{n}$? In other words, what's $Pr[\text{...
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18 views

How to conceptually differentiate “average of frequencies” vs. “total frequency”

I'm running an experiment where I want to see the frequency of saccades (a type of eye movement) in seconds. I've collected data of different trials and I'd like to know of the proper way to ...
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9 views

Why does the mean centered autocorrelation have a slope of -1?

I'm fundamentally not understanding something about the autocorrelation function (as defined by numpy.correlate). Let's say I create a bunch of random signals $s_1, ...
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1answer
31 views

Let $f(x)=2 x^{-3}$ for x between 1 and Infinity, $f(x)=0$ otherwise be the pdf for a random variable $X$, find $F(x)$ [closed]

Let $f(x)=2 x^{-3}$ for x between 1 and Infinity, $f(x)=0$ otherwise be the pdf for a random variable $X$, find $F(x)$. Could you please help me how can I find it ?
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1answer
31 views

Using the Central Limit Theorem to calculate a mean from Poisson distributed random variables

Firstly, I am studying the basic concepts of statistics and so any explanations, advice and suggestions are more than appreciated. Onto the problem- I am given the central limit theorem and understand ...
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1answer
28 views

let $f(x)=(3(x+x^2))/14$ and $x$ between $0$ and $2$ , zero otherwise be the pdf for a random variable $X$ ,Find the median and the mode?

let f(x)=(3(x+x^2))/14 and x between 0 and 2 , zero otherwise be the pdf for a random variable X Find the median and the mode ` Could you please help me Is it correct or not?
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2answers
34 views

Calculating grade averages [closed]

Dear mathstackexchange, There is one module at college that im worried about, can you help me calculate what I need to pass? The homeworks are worth $20\%$ and I got $8\%$. The final is worth $80\%...
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1answer
26 views

Least squares without solution

Talking about simple linear regression (k=1), in which cases the Normal Equations have unique solution? And infinite? And when the Normal equations have no solution?
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0answers
12 views

Calculating the normalizing factor in the VonMises-Fisher distribution on $S^p$

I'm going quickly through the VonMises-Fisher distribution $M$ on $S^p$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $\kappa \...
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Hurst exponent Algorithm not well defined? What to do in this case?

The R/S algorithm is well known for finding the Hurst component of a given data set. I have one query about whether it is well defined? Suppose in a given partition, all of the values were equal to ...
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8 views

Equilibrium states for the Curie-Weiss-Potts model and large deviations.

First of all I am not completely sure if this question should be posted here instead of Physics SE but I'll try. I am doing a project in Large deviation theory applied to statistical mechanics and I ...
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1answer
26 views

Degrees of freedom of t-test in multiple regression .

Formula of t-test in regression is, $ t=\frac{\hat{\beta}-\beta}{se (\hat{\beta})} $ and the degrees of freedom of t-test is (n-k) because we estimate $\hat{\sigma}^2$ from RSS and the RSS has (n-k) ...
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29 views

Probabalistic solve of system of equations

I'm engineer, not mathematician, so excuse me for wrong terminology, but I hope you'll understand the problem. Example situation: I have N electronic components. Each of them has reactance and ...
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21 views

Probability in Chestnut Genetics [closed]

This question concerns cross breeding American and Chinese chestnuts to transfer the blight resistance inherent in Chinese chestnut to the American chestnut. The chestnut genome contains approximately ...
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2answers
31 views

Statistics basics

Given that $X$ has mean $a$ and variance $b$. Then $E(X^2) = a^2 + b^2$. Why is this true? Please provide a proof alongside any other relevant information. Thanks in advance.
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1answer
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$((n-K)s^2)/\sigma^2$ what is this in terms of matrix linear regression?

$$ \frac{(n-K)s^2}{\sigma^2} $$ what is this in terms of matrix linear regression? Has Chi Squared Distribution with (n-K) df
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Notation: should Markov chains steps be noted by uppercase or lowercase letters?

I'm reading the chapter about perfect sampling of the "Monte Carlo Statistical Methods" by Robert and Casella, 2004. I've got an issue about notation, when they talk about random mappings, they say $$...