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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175 views

The Wald test with Poisson distribution

Let $X_1\ldots X_n\sim \operatorname{Poisson}(\lambda)$. Let $\lambda_w>0$ be given, I am trying to find the size $\alpha$ Wald test for $H_0$: $\lambda=\lambda_w$ vs $H_1$: $\lambda\neq \lambda_w$....
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43 views

Show that $Y$ and $Z$ are independent and find their distributions

Suppose that $X\sim\exp(\lambda=1)$. Let $Y$ be the integer part and $Z$ be the fractional part. Show that $Y$ and $Z$ are independent and find their distributions. This one is kinda confusing. Any ...
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31 views

Meaning of $\nabla$ in the Langrange Principle's equations

I was trying to understand this Lagrange Principle, but to fully understand it, I need first to understand the notation. In the Wikipedia's article related to this argument, we have the following $$\...
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34 views

5 distinct number distributed to 5 persons

$5$ distinct numbers are randomly distributed to players numbered $1$ to $5$. Whenever two players compare their numbers, the one with the higher one is the winner. Initially, players 1 and 2 compare ...
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35 views

Fisher Information for Exponential RV

Let $X \sim exp(\lambda_0)$; i.e, an exponential random variable with true parameter $\lambda_0 > 0$. The density is then $f(x;\lambda_0) = \lambda_0 e^{-\lambda_0 x}$. For a given $\lambda > ...
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26 views

Integral $\int_0^s \int_0^t e^{-|p-q|} dp dq $ absolute value of difference between two variables in integrand.

In working out the autocorrelation function of a random variable the following integral needed to be computed but I cannot work out how to do it: $$\int_0^s \int_0^t e^{-|p-q|} dp dq = 2 \min (t,s)+e^{...
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24 views

measuring the regularity of a grid

I'd like to find a method for scoring the difference in regularity of points in grids - there are two examples below (left) a relatively well ordered grid and (right) a relatively disordered grid. ...
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23 views

Probability Density Function (Integration)

A probability density function is given by $$f(x)=\left\{\begin{matrix} ke^{-2x} &x\geq0 \\ 0&otherwise \end{matrix}\right.$$ Find $k$ My attempt, $k\int_{0}^{\infty }e^{-2x}dx=1$ But ...
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20 views

For what value of $k$ is $f(x)=\frac{k}{(1+x^3)}$a distrib. function and what is its variance?

Let $X$ be a random variable with the folowing distribution function: $f(x)=\frac{k}{(1+x^3)}$ for all $x>0$. Find a value for the constant $k$ for which $f$ will be a distribution >...
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41 views

Combinatorics question involving derangements of elements

If $n$ people put their names in a bag, mix it up, and redraw at random, what is the probability that exactly $i$ people get their names back? I have an expression we learned in class about the number ...
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7 views

Bounding the spread of a random variable

I have a random variable $y$ with a finite distribution of values $(y_1,y_2,\dots,y_d)$, with associated probability $(p_1,p_2,\dots ,p_d)$ and average $\langle y \rangle = \sum\limits_{i=1}^d x_i y_i$...
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27 views

Variance of residuals from simple linear regression

I am trying to compute $Var(e_i)$. So far I have $Var(e_i)=Var(y_i-\hat y_i)=Var(y_i)+Var(\hat y_i)-2cov(y_i,\hat y_i)$ Now, I know that $Cov(y_i,\hat y_i)=var(\hat y_i)$ but how do I prove ...
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21 views

How to prove the independency of the two generated Gaussian distribution from Box-Muller method?

Right now, I know the two generated numbers obeys the Gaussian distribution, that, $$ \left\{ \begin{align} P_1=\sqrt{-2\log(U_1)} \cos(2\pi U_2)\\ P_2=\sqrt{-2\log(U_1)} \sin(2\pi U_2) \end{...
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29 views

Odds of one person picking number out of a possible 16, 3 times in a row, with 13 people and from varying positions

My friends and I play a game with sixteen individual numbered balls. Only 13 people are allowed to play at any given time and 13 people have played all 3 games. It takes a 16 to win. The same person ...
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1answer
40 views

Exponential distribution, how to apply in this task?

We have a restaurant, where glasses brake every $6$ month with exponential distribution. What is the probability, that From $5$ glasses, at most $3$ will break in $12$ month? From $500$ glasses, at ...
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17 views

how to find weight of some daily data?

I am collecting some daily data and I want to calculate weight of this data. I'm looking for a math theorem or model or any tip to solve. Let me explain it with an ex: ...
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19 views

Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
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44 views

Calculate sample variance from tallied data

How friends, I am studying Statitics on my own and I am looking for someone to promp me to understand how to approach questions of this nature. I am used to frequency tables but not this one. The ...
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36 views

Continuous Random Variables - Waiting Time Density Function

Suppose you arrive at a public phone booth and find someone else using it. Suppose T is a random variable that represents the waiting time till you are able to get in. Suppose further that the mean ...
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Is there a standard method for converting visits (i.e., visits to the National Parks) to an equivalent resident population?

Purely for my own amusement, I am estimating the violent crime rate in the National Parks. To make sense out of the low number (about $350$ assaults per $275$ million visits, or $1.3$ per million), ...
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11 views

Showing this Random Variable is not complete

Let $0 \leq \theta \leq \frac{1}{3}$ and: $$X = \begin{cases} 1 & \text{w.p. $\theta$} \\ 2 & \text{w.p. $2\theta$} \\ 3 & \text{w.p. $1-3\theta$}\end{cases}$$ If $X$ was complete, then $...
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16 views

Finding the limiting distribution of a bivariate random sample

The question and solution is included below. My main concern is the workings of the calculations; I understand that the main goal of this question is to use the Central Limit Theorem for iid sequences ...
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31 views

Prove that the random variables $X$ and $Y$, $EX^2< \infty$ and $EY^2<\infty$ applies: $DX=E(D(X|Y))+D(E(X|Y))$

Prove that the random variables $X$ and $Y$, $EX^2< \infty$ and $EY^2<\infty$ applies: $$DX=E(D(X|Y))+D(E(X|Y))$$ $(D - \text{ variance }E- \text{ expectation })$ This semestar, we have been ...
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37 views

Condition Expectation with Order Statistics of Uniform Distribution

Consider the family of uniform distribution $U(a,b)$. 1) Prove that $T = (X_{(1)},X_{(n)})$ is sufficient 2) Calculate the conditional expectation $E(\bar{X}|T)$ using the fact that $X_{(2)}...
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57 views

What's the probability of a future polling result falling in a given range?

Question Each day, Gallup polls U.S. Employee Engagement. You can see 7-day rolling averages of the daily numbers here. Assume you have a set of historical daily numbers (ie, ...
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89 views

Laplace distribution moments- what is wrong with this solution?

See the solution to a Laplace distribution moments formula (Answer by Michael Hardy) http://math.stackexchange.com/q/835045 I used this solution as well and I do not see any errors, yet the result ...
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1answer
85 views

What does it mean to “marginalise out” something?

Especially in machine learning one often reads the phrase "to marginalise out" something, and while I understand that this means to integrate over a property, I cannot quite grasp the larger ...
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13 views

Statistical method to prove that variations are not important?

I'm checking the effect a specific substance has in the elongation of the root of variousplants of the Solanum genus. I had my plants grow in soil with different concentration of hormones. My results ...
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23 views

Expectation of a statistic conditional on another statistic

$X_1,\ldots,X_n$ are i.i.d with density $f(x;\theta) = \theta x^{\theta - 1}, 0<x<1$ I'm trying to work out: $$\mathbb{E}(-\log X_1 \mid \sum_{i=1}^n \log X_i = k)$$ How do I go about this? I ...
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32 views

NORMAL DISTRIBUTION(PROBABILITY)

A particular fruit's weights are normally distributed, with a mean of 271 grams and a standard deviation of 22 grams. If you pick one fruit at random, what is the probability that it will weigh ...
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7 views

Using Laspeyres Index to Analyze CAC

I am trying to figure out the best approach to accurately measure customer acquisition cost (CAC) performances for multiple acquisition types when I only have one total spend. To better visualize, ...
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36 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...
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39 views

Which 5 pages did she count the number of errors?

An editor wishes to make a statement about the mean number of errors per page in a fifty page document. Since she does not want to look at every page, she decides that she will take a simple random ...
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28 views

Modify process to semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space. We ...
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10 views

Formula to Depict difference in ratio from Total Spend to Various Acquisition Behaviors

I am trying to come up with a formula that accurately depicts the differences in the customer acquisition cost for various avenues of acquisition. While the standard cost/lead formula works for a ...
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90 views

“Nested” Binomial Distribution

I am currently working with a statistical distribution, and I'm wondering if any exploration has been done on this. The distribution is denoted $\xi$. To construct $\xi$ we use auxillary random ...
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37 views

A Bayesian exercise

I have encountered the following problem in a book I am reading: Suppose you are offered to participate in the following game: Two fair dies are thrown untill '1' will apear (in one of them at ...
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11 views

Design of First order Gauss-Markov model

Consider a First order Gauss-Markov system which is governed by following equation $\underline{x}_{t+1}=\boldsymbol{F}\underline{x}_{t}+w_t$, Where $\underline{x}_{t} \in \mathbb{R} ^{p \times 1} $ ,...
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1answer
66 views

Most powerful size-$\alpha$ test for $f(x|\theta)=\theta x^{\theta-1}$ where $0 < x < 1$

The question is Let $\theta_0 < \theta_1$, show that the most powerful size-$\alpha$ test of $H_0:\theta=\theta_0$ vs. $H_1:\theta=\theta_1$ rejects for small values of $T$. I'm having issues ...
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1answer
29 views

MLE of β in the gamma distribution?

So I have the pdf for the gamma distribution, $$f(x) = \frac{1}{\Gamma(\alpha)} \beta^\alpha x^{\alpha - 1} e^{-\beta x} $$ and I'm having trouble getting to the MLE of $\beta$, which should be $\frac{...
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3answers
64 views

Likelihood of a correct diagnosis of a disease

A certain cancer is found in one person in 5000. If a person does have the disease, in 92% of the cases the diagnostic procedure will show that he or she actually has it. If a person does not have the ...
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37 views

A question about probability theory, convergence in distribution and bounded in probability Thanks!

This is the question,please click it to see the question My question is in the picture above(click the link). if we know (A.10), we want to prove convergence in distribution below, why we need to ...
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141 views

Proving self-consistency of sets of postulates e.g. for spatial Poisson point process

Suppose you have a list of postulates/properties P1, P2, P3 etc. that you would like a (possibly not yet defined) mathematical object to satisfy. Is it possible to prove that the properties P1, P2, ...
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22 views

Conditioning multivariate Gaussian on a function of coordinates

I have a pretty general question and I would really appreciate if you give me any hints or point me towards some relevant literature. Suppose $X$ is an $n$-dimensional Gaussian vector. What is the ...
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1answer
12 views

Strange interval for the random variable $Z_n$ when solving an exercise related to the CLT

I was looking at an exercise which is related to the central limit theorem, but I don't understand a step of the solution, which is when it switches between calculating the probabilities that a sample ...
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1answer
83 views

Conditional Probabilities- Lining up 12 blocks

A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is yellow. The blocks are arranged randomly in a line. If no two black blocks are next to each other (a) what is the ...
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11 views

Find UMVUE for $\frac{1}{p}$ where $X_1,\cdots X_n$ are i.i.d Geometric r.v.s

I know $P(X=x) =p(1-p)^{x-1}$ for a geometric random variable and that $E[X_1]=\frac{1}{p}$. The previous part had me compute the Rao-Blackwell estimator for $p$ which was convenient since $p = P(X=1)$...
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120 views

Sample Space & Combinations/Permutations

This is the questions - "List the sample space for three individuals chosen @ random to vote either Democrat or Republican. List the distinct combinations and permutations" These are what I got for ...
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213 views

Conditional Probabilities- picking balls out of a bag

3 red, 3 green, 3 blue, and 3 orange balls are in a box and 6 of these balls are drawn at random from the box. If 2 of the 6 drawn balls are red and 2 of them are green, what is the probability that ...
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93 views

Estimating Parameters using Method of Moments and Maximum Likelihood and Finding expected values/variance

Let's say we have a dataset $(x_i, Y_i)$ on each randomly n chosen non cities in a country where $x_i$ i=1,...,n is the known population size in city i with cancer. Say $Y_i$ has a Poisson ...