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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1answer
29 views

How to determine a conditional distribution

Consider: $$X\stackrel{d}{=}N(\mu, \sigma^2)\qquad Y\mid X \stackrel{d}{=}N(\alpha+\beta X, \tau^2) \qquad U\mid Y,X \stackrel{d}{=} N(0,\nu^2)$$ Determine the distribution of $W=X+U$ given ...
1
vote
2answers
289 views

Mode from continuous distribution

I've continuous distribution set of values. For example: 0, 0.01, 0.012, 1.2, 5.33, 5.24, 5.38, 30.20, 30.21, 30.13, 30.12 I want to calculate most frequent ...
0
votes
2answers
50 views

Derive the distribution of a lower censored s.v.

I could use some pointers solving this problem: Given a certain s.v. $X$ with cdf $F_x(x)$ and pdf $f_X(x)$. Let s.v. $Y$ be the lower censored of $X$ at $x=b$. Meaning: $$Y = \begin{cases}0 ...
1
vote
1answer
383 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
3
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1answer
37 views

Show that the variance is biased

I am trying to understand the proof that the uncorrected sample variance is biased (given here) $$ \begin{eqnarray} E[S^2] &=& E \left [ \frac{1}{n} \sum \limits_{i=1}^{n} (X_i - \bar ...
4
votes
2answers
69 views

Limit $ \lim\limits_{n\to\infty}\Bigl[\frac{1}{2^{n/2}\Gamma(n/2)} \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{\frac{-t}{2}}\,dt\Bigr]$

Find $ \lim\limits_{n\to\infty}\left[\dfrac{1}{2^{n/2}\Gamma(n/2)} \displaystyle \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{\frac{-t}{2}}\,dt\right]$ This looks like the p.d.f. of a chi-square ...
0
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1answer
49 views

Example of a real-world situation where multivariate analysis is applicable.

I have searched a lot of site to understand the situation where multivariate analysis is applicable. But not got any easily understandable example. Would you please give me a real-world example where ...
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0answers
27 views

4/12 targets are red, 5/18 arrows are red. chance that at least 1 or more red arrows hit a red target.? [closed]

If a target has been hit, the target can't be hit again Obviously not all red arrows can hit a red target so it's more chances that at least 1 or up to 4 of the 5 red arrows will hit a red target
8
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1answer
99 views

Heavy-tailed distributions

I have encountered the following two definitions of heavy-tailedness (right tail) for a $[0,\infty)$-valued random variable $X$ satisfying $\mathbb{E}[X]<\infty$: (i) ...
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1answer
384 views

Expected Value and Standard Deviation for sample

Randomly selecting 50 people from a population, 45% say 'YES' and 55% say 'NO'. Assuming that the true percentage of people in the population who say 'YES' is 48%, what is the expected value and ...
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2answers
42 views

General sufficient condition for independence of these two random Variables.

I need to state and prove a general sufficient condition on(a,b,c) for independence of two random Variables. We have that $a,b$ and $c$ are real numbers and the random variables are below: $$ ...
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0answers
14 views

Is the large-$n$ limit of scatter matrix always equal to the underlying covariance matrix?

While trying to answer this question scatter versus covariance the following occurred to me: Suppose $f(\vec{x})$ is a probability distribution function for which the mean and the covariance matrix ...
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0answers
17 views

Double mass analysis and statistical errors

Double mass analysis The normal field of application for double mass analysis is to find out inconsistencies between sets of data. Say you have two sets of data $x_1,\dots, x_n$ and $y_1,\dots,y_n$. ...
0
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1answer
265 views

UMP of a Beta($\theta,1$) distribution

I need to find the UMP of a BETA$(\theta,1)$ distribution. I know that the pdf of this problem is $$f(x;\theta)=\theta x^{\theta-1}=\theta e^{(\theta-1)\log{x}}$$ Thereby this family contains a MLR ...
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0answers
21 views

Mutiple Regression, calculating R-squared

If I have two regressors in multiple regression equation y=b0 + b1*X1 + b2*X2, how can I find R-squared for the model?I need to know the written formula(not in excel) for two independent variables as ...
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0answers
45 views

Statistics; Hypothesis Question

A die is tossed 120 times with the following result; Number of turned up: 1 2 3 4 5 6 Total Frequency 30 25 18 10 22 15 120 Test the hypothesis that the die is ...
0
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1answer
19 views

relation between covariance matrix and SVD of a given matrix

Suppose there is a given matrix $M$ of $m \times n$ dimensions. What is the relationship (if any) of its covariance matrix and its SVD's singular values matrix? This page says that "the SVD ...
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0answers
31 views

Marginal density from joint distribution of 3 random variables

Suppose random variables X, Y, Z have joint probability density function given as \begin{equation} f_{XYZ}(x,y,z) = \frac{2x}{(A_2-A_1)(R_2^{2} - R_1^{2})y^{2}z} e^{-z^{2}/2} \end{equation} where ...
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0answers
21 views

user specified function in R program [migrated]

I would like to create a function which can be use for different sets of data. This is my function (I make it as source code): ...
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2answers
58 views

Asymptotic Maxwell MLE distribution

Consider i.i.d. random samples $X_1,...,X_n$ from the Maxwell Density: $$ f_\theta(x)=\sqrt{\frac{2}{\pi}}\dfrac{x^2}{\theta^3}e^{-\frac{x^2}{2\theta^2}}I_{(0,\infty)}(x) $$ with $\theta > 0$. ...
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0answers
37 views

PDF of product of 3 random variables

I am trying to find the PDF of S = XY/Z^2, where X,Y,Z are independent with X is uniform [-A1, A2], Z is uniform [B1, B2] and Y positive random variable with any distribution (for example Rayleigh). ...
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1answer
550 views

find marginal probability density function

Suppose random variables X, Y have joint probability density function $f(x, y)$. How do i find the marginal probability density function of X , Y if the support is $$ \begin{cases} 0 < x < 1 \\ ...
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8 views

Gaussian Mixture Models - Probabilistic Modelling

I have been attempting this question on EM-Algorithm for a mixture of 2 Gaussian's for 8 straight hours. I have gotten to the point where I keep repeating myself and expecting a different result. I ...
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0answers
46 views

Rigorously, what is the goal of (machine/statistical) Learning and why is that the goal?

After some time doing machine learning and statistical learning theory, I decided to return to my foundations and make sure that the goal of what I am doing makes sense. First let me define $I(f)$ as ...
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1answer
74 views

Answer for prob. Question?

Three friends Bryce, Chris and Kate each decide to go skiing on the same day without knowing the other two are going as well. Bryce is going to decide between resorts A, B and P. Chris is going to ...
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0answers
16 views

Some sort of Chainrule for measures in statistics [closed]

Lets $X$ be a random variable $X:(\Omega,\mathcal{A})\rightarrow (K,\mathcal{B})$. Suppose that for $T:(K,\mathcal{B})\rightarrow (K',\mathcal{B}')$ (a mapping of the sample space) yields that the ...
2
votes
1answer
300 views

Multiple linear regression with interaction

I'm doing a multiple linear regression with interacting variables. I'll give you an example: $y$=value, $x_1$=material, $x_2$=weight, $x_3$=color $x_1$ and $x_2$ are interacting variables but $x_3$ ...
0
votes
1answer
350 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 ...
3
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2answers
310 views

Blackwell–Girshick equation?

We have the following theorem: Let $N$ be a random variable assuming positive integer values $1, 2, 3,\dots\,$. Let $(X_i)$ be a sequence of independent random variables which are also independent of ...
0
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1answer
28 views

3 students share a room in a dormitory. they have 4 cups, 5 saucers, and 3 teaspoons,

3 students share a room in a dormitory. They have $4$ cups, $5$ saucers, and $3$ teaspoons, all different. In how many ways can they set the table for tea? (each set consists of a cup, saucers, and a ...
2
votes
1answer
429 views

Pearson's Chi Squared / Cochran–Mantel–Haenszel test analog to N-way ANOVA

A test was given to two sets of students, CONTROL and EXPERIMENT, that had question A and question B. I want to know if students who got question A right were more likely to get question B right, and ...
0
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1answer
502 views

Comparing annualised volatility from monthly and annual data

I fear there is a very simple answer to this question and its killing me that I can't see it. I am interested in calculating historical volatility: I have monthly index values starting in Jan 2005 ...
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0answers
43 views

$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
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2answers
27 views

Value of constant k which makes the function $f(x)=\frac{k|x|}{(1+|x|)^4}$ a p.d.f.

Let $f(x)=\dfrac{k|x|}{(1+|x|)^4}$, $-\infty<x<\infty$. Then, what is the value for which f(x) is a probability density function ? f(x) will be a p.d.f. if $\displaystyle ...
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0answers
16 views

A simple statistics question on percentile ranks

Let's say that A scored in the 70th percentile on the SAT. B scored in the 35th percentile. Did B score 50% lower than A (35/70=0.5)? Or did B score 35% lower (70%-35%=35%). I'm getting a little ...
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4answers
90 views

Expected value of $X$ given $X > Y$

I'm doing some research and I'm trying to compute a closed form for $ \mathbb{E}[ X \mid X > Y] $ where $X$, $Y$ are independent normal (but not identical) random variables. Is this known?
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2answers
49 views

Maximum Likelihood Estimation

$Y_1,Y_2,\ldots,Y_n$ are independent random variables, each with probability function $$ P(y)=\binom{y-1}2 p^3(1-p)^{y-3},\quad (y=3,4,5,\ldots). $$ Find the MLE of $p$. I tried rearranging the ...
0
votes
1answer
21 views

Expectation values and variance of a distribution function

We have the distribution function defined for $x\geq0$ $$f(x)=Axe^{-\lambda x}$$ In terms of $\lambda$ I got $A = {\lambda}^2$. And thus got from here $$\mu = \lambda^2\int_0^\infty x^2e^{-\lambda ...
0
votes
1answer
33 views

Bayes estimator from a geometric distribution with a uniform prior

X is a random variable with Ber(p), 0 Y is the number of trials until a success occurs. Assume the prior p is unif(0,1). I have trouble in figuring out the posterior density f(p|Y). With the ...
2
votes
1answer
31 views

Intuition for proof of Slepians Inequality

If z is a centered gaussian random variable and $ x_1 ,x_2 ,..,x_n ,y_1,y_2,..,y_n $ are points in $ \mathbb{R}^{2n} $ satisfying $ |x_i-x_j |_2 \leq |y_i -y_j |_2 \ \ \ \forall i,j \in [n] $ then $ E ...
0
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1answer
21 views

Transformation of a random variable.

In this transformation question, I managed to show the g(w) as given, yet I do not quite understand why the domain of the w is within plus and minus infinity. Shouldn't it be -1< w <1 as we ...
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1answer
43 views

Binomial distribution, when variable isn't x

I've been using the formula $$p(x,N)=\frac{N!}{(\frac{N+x}{2})!(\frac{N-x}{2})!} p^{1/2(N+x)} q^{1/2(N-x)}$$ to determine the probability for a dog who walks in a straight line and can either move ...
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0answers
6 views

method to test homogeneity

Two groups of 10 persons each were asked to rate a product. The rating was done one a Likert scale from 1 to 5. Then the ratings were aggregated (and filtered) in the following way: the number of ...
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1answer
29 views

Is there any known method to fit plane onto sampling data?

For example I have the variables x, y (or higher dimensional data in general) and a probability density distribution p(x,y). I want to approximate p(x,y) as a linear function, a plane in this case, at ...
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0answers
44 views

Chi-square or chi-squared?

The $\chi^2$ test/distribution is referred to as either "chi-square" (more frequently) or else "chi-squared" (less frequently). What is the history behind the name? Footnote 2 in this paper by Peter ...
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0answers
13 views

EM algorithm for spam filtering

there is a slide in "http://stp.lingfil.uu.se/~nivre/master/StatMetLecture4.pdf" about EM algorithm in which there is an example about unsupervised learning of spam email. could anyone explain how the ...
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0answers
17 views

LR-test for exponential distribution

I would like to derive the likelihood ratio test for $H_0: \lambda=\lambda_0$ vs. $H_1:\lambda\not=\lambda_0$ with fixed $\lambda_0>0$ for a sample ...
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0answers
17 views

Non invertible ARMA series

I know if you have a non stationary series it can be explosive in the future so it is quite necesary to prove it doesnt happen and treating to solve it by differenciating. But I dont know very well if ...
2
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0answers
28 views

Does the empirical CDF converge in $L^1$ and $L^2$?

I am somewhat confused with the empirical distribution function: Assume we have $X_1, X_2, ...$ iid real-valued random variables with true distribution $F_0$. Then by the Theorem of ...
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0answers
20 views

Application of Multivariate Analysis

The following situation is proven valuable where multivariate analysis can be applied. This example is taken from the book Applied Multivariate Statistical Analysis ...