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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Determining statistical correlation between XYZ points

I have a set of two 3D points, both 3D points (two points in each set) representing the same object - just in different states (State A, State B). I'd like to see if it's possible to predict the ...
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1answer
40 views

What is the probability that statistical significance will be achieved in a second test?

Assume I'm thinking of investing in a small biotech company. In a phase 2 study with 120 patients split 80/40 between the new drug and old drug, the progression free survival (PFS) rate of the ...
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1answer
16 views

Small question concerning the proof of the unbiased estimate of the population variance

The main purpose of this question is to check my understanding. As in I have an answer that I think is correct, but I'm not sure, since Stats is not my forte. So given a random variable $X$, we take ...
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1answer
59 views

Probability generating function of bivariate Poisson distribution!

Problem setup: $X_1=Y_1+Y_0,X_2=Y_2+Y_0$ where $Y_1, Y_2\text{ and }Y_0$ are independent Poisson random variables with parameters $θ_1, θ_2\text{ and }θ_0$, respectively. I know that the joint ...
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17 views

Goodness of fit and the important of your bins

where can i find articles that have relation of the chi-square test with respect to bin in multidimensional?
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10 views

The Kendall tau distance

In book "Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne" is written The Kendall tau distance between two rankings is the number of pairs that are in different order in the two ...
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1answer
27 views

Continuity Correction Problem.

We're supposed to use continuity correct with a random variable $X$ that can take only integer values , an equivalent expression for : $4\leq10X<48$ is ? Clearly , it should be $0.4\leq X<4.8$ ...
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1answer
47 views

Is there a fundamental way to prove Generalized Mean is a increasing function

Generalized mean: $$M_k=\left(\sum_{i=1}^n\frac{{x_i}^k}{n}\right)^{\frac 1 k}$$ I try to prove $L=\ln\left(M_k\right)$ is increasing. ...
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2answers
19 views

Commenting results in a ratio scale

Consider the following plot: Is it mathematically correct if I say blue is 50% lower than red. Because from the plot it ...
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1answer
444 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
841 views

For the binomial distribution, why does no unbiased estimator exist for $1/p$?

Suppose that $X \sim \mathrm{Binomial}(n,p)$ for $0 < p < 1$ Why does no unbiased estimator exist for $1/p$? My approach: We try to find the structure of $E_p(U(x))$, where $U(x)$ is any ...
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1answer
45 views

Stat: Probability to have one element of a combination identical to one element of another combination

For a business application, I currently have to provide the probability we are going to have an issue in one application. The combination is composed of N unique ...
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2answers
46 views

Regression of Irregular Exponential

I am trying to model the population growth of countries with the following logistic equation: $$p(t) = \frac{P_oK}{P_0+(K-P_0)e^{(-rt)}}\tag{displayed}$$ Where $p$ = population; $P_0$ = initial ...
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1answer
18 views

Estimating variance of estimator of bernoulli process

The maximum likelihood estimate of a Bernoulli process is simply given by $\hat{\theta}=\frac{\sum X_i}{N}$, where N is the total number of bernoulli trial and $X_i$ is the outcome of each trial. ...
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1answer
19 views

Polynomial Fitting of Circular Data Object

This is a very odd question. I have a one dimensional data set that is graphed on a histogram. I am trying to curve fit this data set (using the class midpoints as the x values, and the frequencies as ...
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1answer
37 views

Finding probability

A random sample of size $n = 1460$ is obtained from a population whose size is $N = 1,500,000$ and whose population proportion with a specified characteristic is $p = 0.42$. What is the probability of ...
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2answers
338 views

Variance of a piecewise pdf

How can I calculate the variance of a piecewise continuous function? For example, $f(x)=0.2$, $0 \le x \le 0.5$ ; $2.4x-1$, $0.5 \le x \le 1$
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62 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
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2answers
435 views

Poisson Distribution for Consecutive Figures

I am trying to find the probability for a Poisson distribution. The mean is two cars sold per day. The question is: "What is the probability that at least one car is sold for each of three ...
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1answer
23 views

Find the MLE of $N(\theta,\theta)$

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Find the MLE of $\theta$. I got $\frac{\partial logL(x|\theta)}{\partial ...
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2answers
31 views

Find the posterior of $\theta$.

$\theta \sim \text{ Uniform}(0,1)$ and $X|\theta \sim \text{ Bernoulli}(\theta)$. How would I find the posterior of $\theta$? The likelihood of a Bernoulli is $p^{\sum{x_i}} (1-p)^{n-\sum x_i}$. ...
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4answers
2k views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
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1answer
22 views

Comparing expectation values on different measures

I'm interested to know if it's possible to construct an inequality for the expectation values of a certain function over two different measures, i.e. can we say anything about these two equations? ...
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1answer
32 views

Perform a descriptive statistical analysis and interpret your results.

The following list gives the academic rank of the 25 female faculty members at a small liberal arts college, where L represents lecturer, F represents full professor, A represents assistant professor, ...
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2answers
42 views

Why are there many notations for expected value?

I saw from literature that the expected value of a random variable $f(X)$ is either $E f(X)$, $E(f(X))$ or $E[f(X)]$. Is there a standard which one notation should one use? Is the expected value a ...
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27 views

Closed form solution to the integral of a product of dirichlet distributions

Can we get a closed form solution to the integral of a product of dirichlet distributions? Example: $$D(\vec{x}|\vec{\alpha}) = ...
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1answer
14 views

Prove that, for any (possibly correlated) collection of random variables, $Var(\Sigma_{i=1}^{k} X_i) \leq k \Sigma _{i=1}^{k} Var(X_i)$. [closed]

Prove that, for any (possibly correlated) collection of random variables $X_1, X_2, X_3,...X_k$, $Var(\Sigma_{i=1}^{k} X_i) \leq k \Sigma _{i=1}^{k} Var(X_i)$. And construct an example with $k \geq 2$ ...
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2answers
1k views

Get a Fisher information matrix for linear model with the normal distribution for measurement error?

For given linear model $y = x \beta + \epsilon$, where $\beta$ is a $p$-dimentional column vector, and $\epsilon$ is a measurement error that follows a normal distribution, a FIM is a $p \times p$ ...
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14 views

Some difference in notation in a project report [on hold]

I am focusing on a specific model in my study. When I am writing my project paper, for describing the estimation procedure, can I describe it more generally with some difference in notation than my ...
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1answer
59 views

iff $E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$

Question: $X_n$'s are i.i.d then $$E\Bigl(\sup_{n\geq 1} \frac{|X_n|}{n}\Bigr)<\infty \iff E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$$ My attempt: for $\Rightarrow$ part, because $\limsup ...
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When can we not use Jensen's Inequality to compare risks?

I have two closely related questions regarding Jensen's Inequality. To show that for any estimator $\delta(X)$, there is another estimator based only on sufficient statistic $T$ that gives the ...
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1answer
29 views

Confusion with Chi Squared interpretation

I'm getting confused interpreting the chi squared distribution. I have a number of chi squared values between 10 and ~100 for a number of parameter estimations fitting data to a model and I have 25 ...
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2answers
52 views

Differential Entropy

I'm a little temporarily confused about the concept of differential entropy. It says on wikipedia that the differential entropy of a Gaussian is $\log(\sigma\sqrt{2\pi e})$. However I was thinking as ...
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7 views

reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. Let ...
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1answer
441 views

Proof for Standard Deviation Formula for a Binomial Distribution

I understand the concept of standard deviation as the square root of the square of the mean of each sample value - the mean of the sample values. Here is the mathematical representation (I've solved ...
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1answer
21 views

How to do a significance test without the standard deviation?

I do an experiment twice, each time with 1000 trials. In the first run, I get 450/1000 positives and in the second run, I get 485/1000 positives. Is there any way to test if these results represent a ...
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11 views

Statistical/ML models when observations have different amounts of input

Let's say we're predicting an employee's performance review score for the following year based on his/her performance review scores from each previous year of their employment. We might have these ...
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31 views

Find a function such that follows to normal in distribution

Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each ...
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17 views

estimate the effective size of a gaussian?

How can I estimate the effective size of a gaussian? How bad is think that the size of the gaussian is twice its FWHM? Exist a better way of estimate this effective size?
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1answer
14 views

calculating mean and margin of error

A 95% confidence interval for the mean is given by $(210.4, 213.2)$. Give the mean and margin of error of the sample. The mean would just be $\frac{210.4 + 213.2}{2} = 211.8$. But how would you get ...
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1answer
43 views

Statistics Set Theory [closed]

Prove or disprove the following statement: If $p(a)= p(b)= q$ then $p\left(a \cap b\right) ≤ q^2$ We know nothing know about sets $a$ and $b$. They may or may not be disjoint.
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1answer
70 views

A Wi-Fi password hidden in statistics expression

You can see the image which is very popular in social media currently. We want to know the password before some bad guys attack it :) I think the expression is written using statistical notation. ...
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1answer
22 views

Maximum Likelihood Estimates Question Using Bernoulli

Suppose that $X$ is a discrete random variable with $P(X = 1) = p$ and $P(X = 2) = 1-p$. Three independent observations of $X$ are made: $x_1 =2, x_2 = 1, x_3 = 2$ a.) Write out likelihood as ...
2
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1answer
24 views

Expectation of Independent Variables Equals Zero?

Given $n$ independent random variables, $X_1, X_2, ..., X_n$ , each having a normal distribution, why is it that the following expectation holds? $$E[(X_i - \mu)(X_j - \mu)] = 0$$ where $i \neq j$ ...
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40 views

Integrate the Dirichlet distribution over finite intervals of each random

Is there a close form solution for the integration of the dirichlet distribution over finite intervals of each random variable set up over the simplex. For example $$\int_{a}^{b}\int_{c}^{d} ...
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19 views

Variance decomposition of double sum

This question is about the variance decomposition of a double sum. For a single sum, it is well known that: ...
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17 views

Confusion concerning joint probability function

In the course notes I am studying, the author comments that the joint distribution function $F_{X,Y}$ completely determines the stochastic vector $(X,Y)$ and then proceeds to calculate the following ...
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1answer
18 views

marginal probability density function

Let $X$ and $Y$ be two continuous random variables with joint probability density function $$f_{x,y}(x,y)= \begin{cases} 1/2, & \left\lvert x\right \rvert+\left\lvert y\right \rvert\leq 1\\ 0, ...
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2answers
40 views

Existence of complete sufficient statistics

Suppose $X_1,\ldots,X_n$ are iid r.v.'s, each with pdf $f_{\theta}(x)=\frac{1}{\theta}I\{\theta<x<2\theta\}$. I find the minimal sufficient statistics $(X_{(1)},X_{(n)})$. I am trying to prove ...
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1answer
33 views

Distance between two p.m.fs

I am stuck with the following problem from research. Is there any existing distance measure which can compare two probability mass functions with different support? For eg. for pmfs $p_1$ and $p_2$ ...