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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Loss of information while projecting multidimensional data

I'm interested in the evaluation of the loss of information after projecting multidimensional data. Since the dimensional reduction is a common tool to analyse data,a question about the loss of ...
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probability generating function negative binomial distribution

I want understand this formula,can you help me? suppose we have negative binomial distribution like this: and i want to understand this probability generating function: thanks a lot.
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25 views

Is a least squares method a cost function?

I am kind of confused in the role that a cost function plays in finding the best fit line. I usually find the line of best fit using least squares but the other day I saw an ML lecture online ...
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Confidence intervals and dividing by $n^{0.5}$

I am studying a-level and am confused when I divide the standard deviation by $n^{0.5}$ when doing confidence intervals. To put it another way when I don't divide by it? (where $n$ is the number of ...
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$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find an estimator for $θ$ by the method of moments.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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Show that $ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$ is unbiased estimators of $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Show that $\hat{\theta}_2$ is ...
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Show that for $0<k<1$ $P(k < \frac{Y_{(n)}}{\theta} \le 1) = 1 - k^{cn}$.

The distribution function for a power family distribution is given by $$F(y)=\begin{cases} 0, & y<0\\ \left(\frac{y}{\theta}\right)^\alpha, &0\le y \le \theta \\ 1, ...
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What is the best book to learn statistics?

Right now I'm taking a 3 part course on probability and statistics using Schverish & Degroot Probability and Statistics and it is just not helpful. For the first part, which was on Probability, I ...
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9 views

Probability with replacement

I'm having trouble figuring out this probability problem. I've tried multiplying the probabilities together, but I don't think its right. An urn contains one white and four black balls. If five balls ...
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20 views

Steady state of a $4 \times 4$ transition matrix

Normally I just take $q(M_{m\times n} - I_{m\times n})$ to workout the steady state, but here I have: $$\left(\begin{array}{rrrr} 0 & 0 & .8 & .2 \\ .4 & .6 & 0 & 0 \\ .2 ...
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Limiting probability of Markov chain(Terminology)

If I am asked to find the limiting probability of a Markov chain, what does this pertain to? $\lim \limits_{n \to \infty} P^n$? Where $P$ is the stepping matrix and $n$ is the number of steps. "What ...
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31 views

Show that T is a sufficient statistic

Exponential family: $f(x|\theta) = c(x)d(\theta)\exp[a(\theta)b(x)]$, $T = \sum b(x_i)$. I understand that a sufficient statistic is supposed to help factorize $f(x|\theta)$ into two different ...
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8 views

Kth moment of Y = X^m where X is the Weibull distribution

I am trying to find the $k^{th}$ moment of $ Y = X^m $ where $X$ is the Weibull distribution with parameters $ m, \alpha $. So far I have set up the integration of the moment generating function after ...
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11 views

A Question on the independence of the sample mean and sample variance

The aim of the following question is to show the given random variable follows a student T distribution. Although it seems quite straightforward at the first sight, I am quite confused about the ...
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58 views

Computing sample variance and error using Monte Carlo

Asked to compute estimator for the following function, $$\theta = \int_0^\infty \mathrm{e}^{-x^2}~\mathrm{d}x$$ which can be solved by transforming the limits to 0 to 1 and solving the following ...
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22 views

Scatter plot : Are the two observed data related

I have a regression question that ask to draw the scattered plot graph and then conclude if the two data lists are related. The two data lists are years of people and their cholesterol level. I went ...
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10 views

Derivation of a fixed effects estimator

I've come across parts of a derivation of an estimator in a paper i don't understand. The log likelihood function is where $Y_i=(Y_{i1},...,Y_{iT})'$, and $X_i$ is a $T\times k$ matrix ...
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19 views

meaing of *, # symbol in statistical data

Please refer to the attached graph and explain the meaning of *, *# The above figure is from European Journal of neuroscience. There was no legend or explanation on the symbols.
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38 views

Continuous probability distributions

The following is a question that is likely to be appear on my exam on Friday (tomorrow) but contextualised into a different scenario. I'm having serious difficulty figuring out how to go about ...
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14 views

Basic business statistics

In 2009, the average charge for tax preparation by Hilda was $\mu=187$. Assuming a normal distribution and a standard deviation of $\sigma=20$, what is the probability that a randomly selected ...
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30 views

Why must average only be used with normal distribution?

In a technical job interview I was asked a question that I couldn't quite make sense of. The interviewer was talking about sampling a sound file to make a visual effect (such as a graph that moved to ...
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Uniform distribution unbiased estimator

Let xi be iid observations in a sample from a uniform distribution over [0,θ]. Now I need to estimate θ based on N observations and I want the estimator to be unbiased. I thought about simple ...
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24 views

Standard Deviation of Population from Sample.

I guess I am a little confused. I am doing t test statistics in my class. I think I know but I would love some insight. I am trying to get the estimated standard deviation for the population. These ...
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19 views

Mean of two normal variates

If $X$ and $Y$ are independent standard normal variates find the mean value of the greater of $|X|$ and $|Y|$
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A die is loaded in such away an a probability

A die is loaded in such away that even number is twice as likely to occur as an odd number. Let A be the event that an even number turns up and B be the event that a number divisible by 3 occurs. ...
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Is Expectation Propagation (EP) affected by the prior?

I understand EP by reading Minka's thesis: http://research.microsoft.com/en-us/um/people/minka/papers/ep/minka-ep-uai.pdf I'm trying to apply it to solve a Bayesian inference problem. However, I'm ...
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50 views

Conditional Probability - Bayes' Theorem

The following question is based on conditional probability. I have been told that it requires an application of Bayes' theorem, which I understand only slightly. If possible, could someone explain the ...
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19 views

Distribution of variance estimator (of normal distribution with known mean)

$X_i$ are independently drawn from a normal distribution with mean $\mu$ and variance $\sigma^2$. $$ \hat{\sigma^2} = \frac{1}{n}\sum_{i=1}^{n} (X_i - \mu)^2 $$ I'm trying to get to the distribution ...
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15 views

Transformation technique and the Normal distribution to form Cauchy distribution.

Suppose $W$ and $V$ are independent standard normal random variables. Show that $X=\frac{W}{V}$ has the Cauchy distribution by finding the joint density of $X$ and $Y =V$ and finding the marginal ...
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17 views

Find the probability that you have three of a kind after you pick 5 cards from a regular 52-card deck.

How would I go about finding this? I was thinking that, since this is a permutation, you would have 52!/(52-5!)? But how would that account for 3 of a kind?
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Conditional probability Maths question

The following questions are an excerpt from a set of questions that are to be completed as preparation to my exam on Friday. Myself and friends believe the answer to part 1i to be 1/7 or 0.143 and the ...
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Identification of linear regression function under $\ell_1$-norm criterion.

Consider a linear system \begin{align*} y = \theta^Tx + e, \end{align*} where $x\in\mathbb{R}^n$ and $e\in\mathbb{R}$ are independent Gaussian random variables with distribution $\mathcal{N}(0,I_n)$ ...
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How to interpret the results of a 2 sample KS-test

I have some data which I want to analyze by fitting a function to it. To do that, I have two functions, one being a gaussian, and one the sum of two gaussians. To test the goodness of these fits, I ...
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2answers
24 views

Finding pdf's from a equation?

Please can someone expalin how i do question 8i in this picture, I have tried finding X in terms of theta but don't know then how to turn this into a pdf (if that is even what it is looking for)? I ...
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64 views

Derive minimum length confidence bounds for a F distribution variance …

Derive minimum length confidence bounds for a F distribution variance $\sigma^2$ and the ratio of two F distribution population variances $\frac{\sigma_1^2}{\sigma_2^2}$. What I got so far is $$ ...
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21 views

sum of two normal distributed random variables

Consider $Z = \frac{1}{2}(X+Y)$ where X and Y are normal distributed variables. It is easy to show that the variance of $Z$ is $1/2$ and the mean is zero. I'm stuck because I want to show this using ...
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3answers
42 views

Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$

Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$. (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$ (b) Can you propose ...
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61 views

LR Test for Exponential Family of Distributions

LR Test for Exponential Family of Distributions: The exponential family of distributions, both discrete and continuous, based on a parameter θ is defined by: f (x |theta) = ...
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37 views

What exactly are order statistics?

I'm very confused by what these are. My definition reads: Let $Z_1,\dots,Z_n$ take the values of random variables $X_1,\dots, X_n,$ arranged in increasing order so that $Z_1\leq \cdots\leq Z_n$. We ...
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expected distance

Let P be a point chosen at random from the unit disk. The unit disk is the disk of radius 1 centered at the origin. It is described by the condition $x^2 + y^2 \le 1$. "At random" means "according ...
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63 views

Show that the likelihood ratio test can be distributed Chi-Squared

Show that the asymptotic likelihood ratio test statistic, Chi-Square LRT = -2log(Λ), to test H0: μ = μ0 vs. HA: μ ≠ μ0 is truly Chi-Squared (df=1)-distributed for Y1,…,Yn~ N(μ,σ^2) when σ^2 is known ...
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1answer
24 views

random number generator statistics

A random number generator has produced the following sample from the uniform distribution on $(0,1): Y_1=.12, Y_2=.87, Y_3=.61.$ Use this and the recipe in (b) of the Special Transformation Theorem to ...
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+50

Website (A/B or Multivariate) testing, Statistical significance not lining up with sample size.

So I have setup testing code on a website, our conversion data is blocked per day and fed into R. This is the R code that I am using to calculate the "confidence" ...
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What is the Cardinal Theorem of Statisics?

I'm doing Straitigraphy (a 3rd year module at university) and my professor asked us to do a "mini" assignment for tomorrow. Max 1 - 2 paragraphs. His topic was "Cardinal Theorem of Statistics." I've ...
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Probability applied to economics

The following two questions are based on the wondrous Statistical topic probability. After attempting both questions I have yet to answer either correctly. If anyone has encountered similar problems ...
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Ancillary statistics and the use of Basu theorem

Let Y1, Y2, ....., Yn be the order statistics of a random sample of size n from the distribution exp(theta) If R= ( n Y1 / M) where M = summation Yi . To show that R and M are independent ?? ...
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29 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let ...
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linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
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42 views

A point chosen at random from a disc.

Ive been working on this question and have managed to complete parts $(i)->(iii)$ but am struggling with the last two parts. For $(iv)$ I end up getting this when trying to find the distribution ...
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74 views

Normally distributed sum of values

Mike's bowling scores are normally distributed with mean 110 and standard deviation 13, while Jack's scores are normally distributed with mean 135 and standard deviation 10. If Mike and Jack each ...