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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Check for Independence

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...
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76 views

What's the “ridge” in Ridge Regression?

In normal least squares, we try to find $\hat\beta$ which minimizes $$\|y-X\beta\|^2$$ Ridge regression expands this to "penalize" certain values of $\beta$ via a matrix $\Gamma$: ...
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145 views

confidence intervals additive?

Suppose that I have N different businesses running during the summer. Let us further suppose that I estimate (using regression) the income for each business with a 95% confidence interval. I then sum ...
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75 views

Generalized Bayes Estimator

Consider a decision problem in which the model parameter, $\theta$, is any integer, the distribution for the integer observation, y, given $\theta$ is $P(y|\theta) = 1/3$ if $y \in [\theta - 1, \theta ...
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48 views

Random var. Y with pdf $f_Y(y) = 4y^3$. Show that $-2\ln (Y^4)$ ~ $X_{(2)}^2$.

Let Y be a random variable which has pdf $$f_Y(y) = \begin{cases}4y^3, & 0 < y < 1, \\ 0, &\text{elsewhere}.\end{cases}$$ Show that $-2 \ln (Y^4)$ ~ $X_{(2)}^2$. Could anyone get me ...
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226 views

Help for $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom.

If $T$ has a $t$ distribution with $v$ degrees of freedom, then $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom. First, I set $$T = \frac{Z}{\sqrt{W/v}}$$ ...
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88 views

Upper and Lower one sided confidence level

Iam trying to calculate upper and lower confidence levels for a parameter, but i can't get it straight (in this case $\sigma^2$): the reference variable: $R_{\sigma^2} := \frac{n-1s^2}{\sigma^2} \sim ...
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534 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $$f(x;\theta) = \begin{cases} 1 & \text{if } \theta-1/2<x< \theta+1/2 \\ 0 & \text{otherwise} \end{cases}$$ Let $Y_1=\min \lbrace X_1,\ldots,X_n ...
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531 views

what the central limit theorem says

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
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169 views

tail probabilities for the sum of independent Laplace random variables

How might I find tail probabilities (pr X>x), or a reasonable approximation, for a variable that is the sum of independent Laplace random variables?
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77 views

Transition from parametric to nonparametric statistics: what is $\Theta$?

During my first statistics course I learned that a statistical model is a collection of probability measures $\mathcal{P}$, where we can index each measure by a 'parameter' $\theta$ such that ...
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288 views

Checking whether a maximum likelihood estimator is biased

So I have a Poisson distribution: $ V \sim \operatorname{Po} \left({\rho v}\right) $ and I've calculated the maximum likelihood estimator $ \widehat{\rho} = \dfrac{\overline{v}}{v} $ from ...
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45 views

correlation of conditional mean function and its parts

The random variables Y and X have a bivariate normal distribution with means 1 and 2, variances 2 and 5 and covariance 3. Compute the squared correlation between y and E[y|x].
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161 views

Wrong result from LLR using Dunning Entropy method

I'm trying to use Dunning's method of calculating LLR to compare word instances between two fulltext indexes. His method uses entropy as part of the calculation. Dunning's blog post: ...
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5k views

Shifted Exponential Distribution and MLE

I was doing my homework and the following problem came up! We have the CDF of an exponential distribution that is shifted $L$ units where $L>0$ and $x>=L$. The CDF is: $$1-e^{-\lambda(x-L)}$$ ...
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3answers
5k views

A bowl contains 10 red balls and 10 blue balls, A women selects ball at random without looking?

How can we solve this question ? A bowl contains $10$ red balls and $10$ blue balls, and a women picks up balls from the bowl, at random, without looking. A) How many balls must she pickup in ...
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35 views

How long does it take

Assume you have a repeating Job, s single Job takes 1 min execution time and you will succeed with 98 % doing this job correct. Your complete Task is to repeat this Job n times, no error is ...
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33 views

Derivation of $ \frac{n-1}{n}\sigma $

Near the last step for deriving the unbiasness check of $\hat{\sigma}$, we come to a point where $$ = \frac{1}{n} \bigg( \sum_{i=1}^{n} \sigma + \mu^2 -n (\frac{\sigma}{n} + \mu^2 ) \bigg) $$ ...
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971 views

Correlation between complex random variables

I am struggling to find the correlation between two complex r.vs; X and 1/Y i.e. E{X*/Y}, where '*' denotes the conjugation operator. The complex r.s X and Y are correlated with each other with known ...
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3answers
278 views

Binomial distributions: Probability that player $A$ wins $x$ games

I think this is a binomial question because it involves success/failure (win/lose) but it doesn't give that information. I'm finding this question difficult because it asks for the probability of ...
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441 views

How to find the Fisher Information of a function of the MLE of a Geometric (p) distribution?

Suppose that we have $X_1,...,X_n$ iid observations from a Geometric($p$) distribution. I found that the MLE of p is $\hat{p} = \frac{n}{n+\sum{X_i}}$. I am trying to find the MLE of $\theta = ...
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137 views

MLE of uniform distribution

Now before I begin, I know this question has been asked multiple times but all the answers but I had so many questions of my own that I figured I should make a new question as my thoughts are ...
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2answers
7k views

In how many ways can 20 identical balls be distributed into 4 distinct boxes subject?

I was practicing math exercises on text book and i got stuck in this question ? ...
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91 views

Testing hypothesis - type I and type II error

I have given following information: $n=15$ - sample size $S_n=0.013$ - sample standard deviation I have to test null hypothesis $H_0(\sigma=0.01)$ agaist alternative hypothesis ...
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111 views

Introductory Statistics/Probability Reference Request

I am a PhD student in mathematics with a background in Pure Mathematics. As such I have pretty much zero background in statistics or anything other than the most basic probability (i.e., what one ...
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54 views

Which probability in this hypothesis test?

We have a hypothesis A (null hypothesis) such that $p\le 0.6$ and B such that $p>0.6$. Now we want to develop a deterministic test $\phi$ for 20 people that has a safety of 95%. Hence we would be ...
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172 views

Statistics - Lost with this question

I'm having trouble doing this question because I don't know where to begin. Could someone walk me through this slowly so that I understand the thought process and how to approach questions like this? ...
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2k views

PCA using SVD in Matlab, a few questions.

I have X = [25, 2000] i.e. 25 subjects and 2000 values (i.e. each subject has a spectrogram that is reduced to 2000 values). My goal is to reduce from 25 subjects to 1 or 2 "subjects" that best ...
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21 views

Does the order of samples taken affect the probability of getting a certain value?

The question is asking what is the probability of taking two samples and having the first one being shorter than 20 and the second one being larger than 25 based on the given density curve The ...
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55 views

Convergence rate of an estimator

Say we are interested in estimating some unknown real scalar parameter $\alpha$ using data. Suppose the estimator $\widehat \alpha_N$ of $\alpha$ using the data is consistent. I want to know what it ...
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57 views

Why is the marginalized inverse-Wishart distribution not equal to the inverse-gamma distribution?

Given that the inverse-gamma distribution is the one-dimensional version of the inverse-Wishart distribution, why will (philosophically speaking) an inverse-Wishart distribution that originally has ...
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58 views

Moment generation function for continuous random variable

Apologies, my math skills have become rusty and I'm having trouble with this question: $X$ is a continuous random variable with $$f(x)=\cases{e^{-(x+2)}&\text{if } -2 < x < ...
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116 views

Solving a Nobel Prize problem won by Dr. Kahneman. Please help.

A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. About 50% of all babies are boys. ...
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1answer
470 views

Showing that the MLE of $f(x\mid\theta)=\theta x^{\theta-1}$ is consistent.

Suppose $X_1,\ldots,X_n$ are IID RV with distribution $$f(x\mid\theta)=\theta x^{\theta-1}, \;\;0<x<1,\,\,0<\theta<\infty.$$ I've found the MLE $\hat{\theta}$ ...
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128 views

Chances of getting mono from beer pong?

In a friendly game of Beer pong there is 1 cup that has been previously used by a person with mono. Assuming the cup is infected what are the chances that you drink the infected cup if you drink 4 ...
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229 views

Fitting a simple linear regression

A professor in the School of Business in a university polled a dozen colleagues about the number of professional meetings they attended in the past five years $x$ and the number of papers they ...
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174 views

Calculating p-value when tcalc, and df are given. Please help.

In a two-tailed t-test for means equality with df= 22, tcalc=3.511, and .001 < p<.01. How to calcultate exact p-value? There was no table or anything else attached to this problem, and I can't ...
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54 views

Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
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251 views

Seasonal adjustment and Fourier analysis

I've been reading up on seasonal adjustment (removing "seasonal" periodic components from a time series) recently and although I see a lot of fancy work around ARIMA models and fancy ways to detect ...
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186 views

Harmonic F statistic

i am interested what does mean Harmonic F statistic in mathematical language?i have search about $F$ statistic and found a lot of explanation,for example like this "**F Statistic The F statistic ...
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48 views

How does the Kurtosis test for normality work?

For example, the skewness test statistic is based on averaging the x^3 of the data. If the distribution is symmetric, there will be similar number of positive x’s and negative x’s, thus x^3 and (-x)^3 ...
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281 views

Finding the expected value of poisson distribution

The number of breakdowns per week for a type of computer is a random variable Y having a Poisson distribution with mean μ. A random sample Y1; Y2, .... Yn of observations on the weekly number of ...
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51 views

Find mean and covariance of transitional distribution from prior to posterior

suppose $p(\mathbf{x}) \sim{}N(\mathbf{x}|\mathbf{m},\mathbf{P})$ is the prior density, where $\mathbf{m}$ is mean, and $\mathbf{P}$ is covariance. ...
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130 views

Can I do reverse curve fitting operation of 2nd order curve y=a+bx+cx^2?

I have data points of x[i] and y[i] where x is independent variable.And,I can model these points as nth order polynomial using any regression analysis.I am doing 2nd order polynomial curve fitting ...
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48 views

Finding maximum likelihood estimates of parameters of a proportional hazards model.

Consider the proportional hazards model $h(t,x)=\lambda(t)\exp\{\beta'x\}$. Let $\Lambda(t)=\int_{-\infty}^t\lambda(u)du=\exp\{\alpha t\}$. Suppose there are $n$ uncensored and $m$ censored ...
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76 views

Estimate correlation coefficient of unknown variable

Consider variable y depends on variable x and z linearly. I have $100$ sample values of $y$ and corresponding $x$ but don't have any values of $z$. The functional model is $$y = \alpha_1x + \alpha_2z ...
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18 views

How do I design a reward function given two parameters

We want to design a reward function R(x, y) where $x \in [0, 1]$ and $y \in (0, 10000)$ such that x is the dominant term when its value is close to or equal to 1, whereas y is dominant when x is ...
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42 views

Linear Regression Question (Linear Algebra) Help!!

Hey guys, I have a quick question. I am trying to prove that the squared sample correlation between fitted and observed values is equal to $R^2$ (coefficient of determination). I am having a lot of ...
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76 views

How to get 2 percentages to a 100%

First of im new to this site and I've never been the sharpest at math, im a web developer by trade. My question is math related but ill just give you a quick background about my website so that you ...
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1answer
2k views

Determine the Asymptotic Distribution of the Method of Moments Estimator of $\theta$, $\tilde{\theta}$

I am having difficulty understanding what it means to find the asymptotic distribution of a statistic. I have the correct answer (as far as I know), but I am unconvinced that I understand the process ...