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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Probability of Getting a “Perfect Score” in the Card Matching Game Concentration

A person is playing the card matching game concentration. There are 40 cards, 20 pairs total. All the cards are shuffled and placed at random face down. A turn consists of two moves and a move is ...
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1answer
16 views

Unclear about the definition of payoff in a mathematical statistics context

I highlighted the area of confusion. The author uses payoff without defining it and I'm not sure where this is coming from or what it means. I am not sure why the following is true - "Moreover, if ...
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1answer
31 views

Linear regression as $\dim(\beta) \rightarrow \infty$

Consider the linear regression, $$ Y_i = X_i\beta + U_i \qquad E[X_i'U_i]=0 $$ where $X_i=(1,W_{i},W_{i}^2,..\ldots,W_i^K)$ and $\beta \in \mathbb{R}^{K+1}$. The joint distribution of $(X_i,Y_i)$ is ...
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1answer
19 views

What test should I use to statistically compare two intraclass correlation coefficients (ICC)?

I need to compare two generalizability (G) coefficients for data that are from two separate populations. G coefficients are a type of intraclass correlation coefficient (ICC). The literature on ...
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1answer
14 views

Should we convert a grouped distribution to continuous if we have to find the median?

Today, in a math exam, I had to solve this question: In the following distribution, write the upper limit of the median class: $$\begin{array}{|c|c|}\hline ...
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1answer
18 views

Can't understand the parameters in adjusted R squared

I suppose this is a really stupid question, so please excuse me, but I can't get it. I'm using Simple Linear Regression (Least Squares) to find a line, fitting some points. The points are actually ...
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1answer
268 views

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms…

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a) What is the ...
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1answer
22 views

estimation problem for two-parameter weibull distribution

Suppose the two-parameter Weibull distribution is given by the pdf $$ f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\}, $$ where $x,a,b>0$. I am ...
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2answers
33 views

Dice Rolling 4d10 with a twist

Suppose I roll two 10-sided dice, 1 die has numbers o, 10, 20, 30 etc to 90. The second die has numbers 0, 1 ,2 etc to 9. These dice are used to create a number from 1 to 100 - example: the first ...
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1answer
22 views

Proving variance of U-statistics is decreasing

I read Wassily Hoeffding's paper "a class of statistics with asymptotically normal distribution". In proving "$n\sigma^{2}(U_{n})$ is decreasing in n" in Theorem 5.2, it simply says "using (5.33) and ...
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4answers
98 views

If $X \sim N(0,1)$, why is $E(X^2)=1$?

If $X$ is a normally distributed with mean $0$ and variance $1$, expectation of $X$ equals $0$ but why is $E(X^2)=1$?
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61 views

Conceptually: A set whose elements can only be probabilistically characterized?

Sorry for the informality here, but was musing over the basic concepts around describing a set in real world usage: A finite set of explicitly named elements, this apple and that apple, nothing more ...
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2answers
15 views

Second moment of chi squared distribution

I've got difficulties in computing the second momentum of chi squared. Chi squared distribution with $n$ degrees of freedom is the sum of $n$ independent distributions $X^2$, where $X \sim N(0;1)$. ...
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1answer
218 views

Gaussian Curve Fitting - Parameter Estimation

I was redirected here because someone in SO pointed out this is more of a math question than a programming question: I have to fit a Gaussian curve to a noisy set of data and then take it's FWHM for ...
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1answer
26 views

Distribution of reversed k-th order statistics

Let $X_1,...X_n$ be i.i.d. Let $Y_{(i)}$ the $i$-th order statistic of that sample. The distribution function of the order statistic is given by $$F_{Y_{(i)}}(y) = \sum_{k=i}^n \binom{n}{k} y^k ...
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1answer
22 views

Advantage of Bootstrapping Confidence Intervals over Standard Error [closed]

I want to compare means from samples of varying size, plug them into a bar graph and have appropriate values to plug in for the error bars. My sample sizes are: ...
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1answer
14 views

Building histogram of latency using sample of input data

Assume I have input data set consisting of a web page response time. I'd like to build histogram from input data, but for practical reasons I can only use sample of data. Based on histogram I want to ...
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0answers
42 views

Special case of Kullback-Leibler additivity

I have three random variables $X,Y,Z$. If $(X,Z)$ are an independent pair and $(Y,Z)$ are an independent pair, then the additive property of the Kullback-Leibler divergence says $K(X,Z|Y,Z) = K(X|Y) ...
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1answer
20 views

Likelihood function of a Poisson process

Fix a window of time $[0,T]$ and say that we get $n$ arrival times in the window from a homogeneous Poisson process. The maximum likelihood estimate (MLE) is just $n/T$ I believe. But what is the ...
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0answers
40 views

What is the “Logistic Regression”? I cannot have a unified concept.

I have a question about logistic regression. Recently, I interested in logistic regression for modeling some classification problem. So I tried to study logistic regression with two books, "Discrete ...
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1answer
23 views

How do I convert non-normal distribution to a normal distribution?

I have the below graph I drew using histogram in Excel but when I got my data and graphed it, certainly it is not a normal distribution. My assignment required me to gather some data and draw normal ...
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1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
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1answer
266 views

Balls in a box probabilities

A box is filled out by 1000 balls. the box can be thought as containing V sites and V balls, with V=1000. The box is repeatedly shaken, so that each ball has enough time to visit all 1000 sites. The ...
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0answers
18 views

Confidence size and coverage probability in a confidence set?

Let $\theta \in \Theta \subseteq \mathbb{R^d}$ be the parameter of interest and let $\theta_0$ be the true population parameter value. Let $n$ be the sample size. Let $CS_n$ be the confidence set ...
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1answer
32 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
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4answers
124 views

Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
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0answers
22 views

Combination of X1.y1 + x2, y2

Thanks in advance. Request to provide your support to solve the problem. I Have a set of X and Y combinations as follows: ...
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2answers
34 views

Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of X

Let X be a random variable with a continuous and strictly increasing c.d.f. function F (so that the quantile function F^−1 is well-defined). Define a new random variable Y by Y = F(X). Show that Y has a ...
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0answers
20 views

Statistics on a column of values

I have a class project that I am working on as an engineering student but the course is probability and statistics so I don't really know how to make sense or use of what I am learning so I thought of ...
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3answers
43 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
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1answer
64 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
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1answer
20 views

Plotting Normal Distribution using Excel

I was trying to experiment some stuff (scaling issues and hypothesis testing) with normal distribution. While doing so, I found out that : NORM.S.DIST(0, FALSE), which takes Z-value, returns prob. ...
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0answers
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What is $f(x|y<\bar{y})$ equal to?? [duplicate]

Where $x$ and $y$ follow a bivariate normal distribution, $x$ and $y$ are not independent, and $f()$ is a probability density function.
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1answer
517 views

Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
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1answer
3k views

How to calculate the covariance matrix

I tried searching a lot on the net and got the following sources: Source One Source Two The first source seems to be incorrect cause when I calculate it using matlab it comes to be different from ...
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1answer
34 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
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3answers
2k views

Calculate variance from a stream of sample values

I'd like to calculate a standard deviation for a very large (but known) number of sample values, with the highest accuracy possible. The number of samples is larger than can be efficiently stored in ...
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1answer
36 views

Differentiating Integrals

This problem appears as example 2d of Chapter 5 in "A First Course in Probability - Ross, 8th ed." Suppose that if you are s minutes early for an appointment, then you incur the cost cs, and if you ...
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1answer
47 views

The regression analysis below relates US annual energy consumption in trillions of BTUs to

The regression analysis below relates US annual energy consumption in trillions of BTUs to the independent variable "US Gross Domestic Product (GDP) in trillions of dollars." 

Which of the following ...
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1answer
37 views

measure of dependence for copula

I have some question about the paper of Schweizer and Wolff (1981). The question concerns about the following bound $$\int_0^1\int_0^1|C(u,v)-uv|\,du\,dv\leq\frac{1}{12}$$ where $C$ is any copula. ...
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2answers
391 views

Let A, B and C be events such that A and B are independent, B and C are mutually exclusive, P{A} = 1/4, P{B} = 1/6, and P{C} = 1/2.

I need to find P{(A ∩ B)'∪ C}. I know P(A ∩ B)' = P(A) + P(B) - P(A ∩ B) + P(C) =1/4 + 1/6 - 1/24 + 1/2 = 21/24 But how do I do the union C part? C is disjoint from everything else, so If I add 1/2 ...
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1answer
27 views

Alternatives to absolute error?

Let me explain my scenario in which I need to calculate absolute error. Lets say the X is the actual value. And X' is the value of X with some error 'e'. So X' = X + e'. Lets say i = 1 to 10000. I ...
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1answer
16 views

“Approximation” of a maximum likelihood confidence set

I have some trouble constructing the "approximate" set, how can it be defined or calculated? Suppose that three characteristics in a large population can be observed according to the following ...
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1answer
40 views

Stratified random sampling without replacement

I came across this statement and can't decide if it's true or false. Statement: In a stratified random sampling without replacement, with proportional allocation to the population size, the sample ...
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1answer
26 views

A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
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2answers
47 views

Limiting distribution.

Let $Y_n \sim \chi^2(n) $. Find the limiting distribution, $(Y_n-n)/ \sqrt{2n}$ as $n\rightarrow \infty $, using moment generating functions. I don't know how to properly calculate the moment ...
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5answers
2k views

The sum of n independent normal random variables.

How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean ...
2
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2answers
55 views

unbiased estimator in a random sample

I Have a statistic statement here which I need to decide if it's true or false Statement: "When the sample size is random, there is no way to get an unbiased estimator for the population average." ...
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2answers
909 views

What is a direct correlation?

I have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the ...
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1answer
73 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...