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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Total boundness of Lipschitz densities

In Devroye and Lugosi 1990 they claim that Lipschitz densities with Lipschitz constant bounded by some $C$ are a closed set and are as a class of densities totally bounded. Why is this? I cannot ...
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1answer
50 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
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1answer
18 views

$\overline{\theta}$ the maximum likelihood estimator of $\theta \implies$?

I can't understand how the following statement holds without any extra conditions on the function $g$: $\overline{\lambda}$ the maximum likelihood estimator of parameter $\lambda \implies ...
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2answers
35 views

Expected area of a random triangle with fixed perimeter

I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle ...
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21 views

convergence of types

I have the following assumptions : $\lim_{t \to \infty} F(a(t)x+b(t))=S(x)\ \forall x \in C(S)$ $\lim_{t \to \infty} F(\tilde{a}(t)x+\tilde{b}(t))=S^*(x)\ \forall x \in C(S)$ and want to show that ...
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13 views

Relation between RSD and Mean

Is there any relation between relative standard deviation and mean? In my data there is 3 mean values like 98.1,99.5,97.8 and relative standard deviation is 0.6,0.7,0.8 respectively. Which one is ...
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1answer
20 views

Is difference between two sets of measurements significant?

Consider the following experimental setting: I have two machines $m_0$ and $m_1$ of which I would like to know which one performs better. For this I have set up an experiment to measure the time it ...
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12 views

For a normal distribution, find the percentage of data that are between 3.5 standard deviations below the mean and 3 standard deviations above [on hold]

for a normal distribution find the percentage of data that are between 3.5 standard deviations below the mean and 3 standard deviations above the mean
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17 views

How to estimate $\sum_{x=1:n}{xf(x)}$ having $\tilde{f}$

I have an estimator $\tilde{f}(x)$ whose error is at most $\epsilon$, i.e., $\frac{|f(x)-\tilde{f}(x)|}{|f(x)|} \leq \epsilon$. I want to estimate $\sum_{i=1:n}i.f(i)$ with a small error. But if I ...
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1answer
26 views

How to get a good approximation or exact value to the average of a group of numbers without storing the numbers in memory?

I am not sure if this is possible, plus I am running on no sleep, so here is what I am asking. Assume I have a list of random numbers. In this instance we will have a list containing $[5, 4, 3, 2, ...
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12 views

Proof of Hammersley and Clifford theorem in Besag's paper

I am reading Besag's paper on Spatial Interaction and the Statistical Analysis of Lattice Systems, see http://www.cise.ufl.edu/~anand/fa11/Besag_Spatial_interaction.pdf. In section 3, it introduces ...
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6 views

Spatially model

Can someone explain to me what a 'spatially lagged autoregressive model' is ? I came across this 'model' by searching new techniques for modeling geographical data.
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14 views

Estimation of real estate

I'm working on a project on estimating the price value of real estate. First I have collected a lot of data (500 000 instances) with details such as postal code, number of bedrooms, build year, ...
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0answers
20 views

Continuous time markov chains, is this step by step example correct

I have some questions regarding CTMC... and most importantly whether the step-by-step example I provide below is correct. My main sources about CTMC are: ([1], and [2]). Let's assume 3 possible ...
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1answer
23 views

poisitive definite matrix imply matrix of full rank? [on hold]

Is a positive definite matrix always of full rank? while is given that the characteristic roots are pisitive
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1answer
25 views

Total variation inequality [on hold]

In an article I've stumbled upon this inequality: $$ V(f) \leq C(f)s(f) $$ where $f$ is a Lipschitz$(1)$ density with Lipschitz constant $C(f)<\infty$ and support $s(f)$. How can this be derived? ...
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17 views

L1 error for scale/translation classes

This is an example given in the article about testability (Devroye and Lugosi 1990). First will introduce my problem, given that we have a density class $\mathcal{F}$ that consists of density ...
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22 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1990). They use some properties of the essential supremum I cannot find. First we have to assume that we have a class of density ...
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1answer
30 views

Statistics - Finding the median

Problem : x = 0-4 , 4-10 , 10-18 , 18-30 , 30-40 f(x) = 15 , 35 , 20 , 20 , 10 Finding the median? This is that I did : Cumulative frequency distribution : F(x) = 15 , 50 , 70 , 90 , 100 ...
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1answer
44 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
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0answers
9 views

What is the distribution of sum of a Rayleigh and a Log-logistic distributed independent r.v.?

Let Z = X + Y. X follows the Rayleigh distribution given by \begin{equation} f_X(x;\sigma_1) = \frac{x}{\sigma_1^2}\exp\left(-\frac{x^2}{2\sigma_1^2}\right), \; x \geq 0 \label{RayleighPDF} ...
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2answers
17 views

Calculating conditional lottery probabilities - and example from DeGroot.

I'm trying to understand this example in Probability and Statistics in DeGroot: http://imgur.com/yjr1vLQ "You learned that the event B = {one of the numbers drawn is 15} has occurred. You want to ...
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1answer
16 views

Work out if the relationship between 2 datasets is constant

I have 2 one-dimensional datasets, let's call them a and b. I want to know the correlation between ...
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1answer
16 views

Biased MLE estimate of mean (expectation)

Please give an example of p.m.f. or p.d.f. , the maximum likely-hood estimate of whose mean (expectation) is a biased estimator . Thanks
2
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1answer
27 views

Mean of this experiment

Let the following experience : Let $x=n_0\in\mathbb{N}$. Each step, $x$ has a $1/2$ probability of increasing by one and a $1/2$ probability of decreasing by one. The goal is to ...
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1answer
14 views

Similar to Kruskal-Wallis Test for multidimensional vector

I need to do the similar kind of test as Kruskal-Wallis Test but for the multidimensional vector. Would you please kindly suggest such kind of test or provide some references for N-d case. Thank you ...
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17 views

solve equation involving digamma function

I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast ...
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0answers
25 views

Inference Two Sample

During the regular season of the NBA, the great Larry Bird made 3,960 free throws out of 4,471 attempts for a career free throw percentage of 88.6%. As of March 21/11, British Columbia’s own Steve ...
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1answer
22 views

Probability with two uniformly distributed cost

I have two uniformly distributed costs that are random and statistically independent. They are 3,000 to 6,000 and 3,000 to 12,000. I am trying to find the probability that the total cost of these two ...
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18 views

Mann-Witney test statistic?

For a Mann-Witney test do we use the T value from the smallest sample size as the test statistic or the smallest T value? I presume it is the former since this is how it is with the wilcoxon rank sum ...
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2answers
19 views

Variance of a dice roll

I am currently working on a problem and am unsure if I approached it correctly. Here it is: ...
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1answer
59 views

Is my coin fair — only 11 trials

My task was to flip my penny $11$ times and record the number of successes (heads) and then determine with $99\%$ confidence whether the coin is fair or foul. my $\hat{p}$ is $0.45$, thus $\hat{q}$ ...
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14 views

Business Stats Question Please Help

For students taking online courses at ABC University, grades have been distributed as follows: A – 40%, B – 30%, C – 15%, D – 10% and F – 5%. A sample of 173 students taking online courses at KSU has ...
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2answers
12 views

Determining level of significance when hypothesis is an interval

For a normally distributed sample: σ = 60 Sample size = n = 12 Sample mean = x = 3450 Null Hypothesis = H_o ≠ 3500 Hypothesis = H_1 = 3500 I need to determine the smallest significance level at ...
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1answer
24 views

What can be said about usefulness of two regression lines?

How one can say that the regression equation $y$ on $x$ is more useful than the regression equation $x$ on $y$ as a predicting formula?
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1answer
28 views

Stats Normal distribution [on hold]

A company pays it employees an average wage of $\$15.90$ an hour with standard deviation $\$1.50$ per hour. Assume the wages are approximately normally distributed. a) What proportion of employees ...
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45 views

A city wants to encourage downtown

could you please help me with this ( part d ) A city wants to encourage downtown employees to use public transportation. Below is the time in minutes to get to work on one morning according to ...
0
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1answer
44 views

Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula. In this way: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the ...
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0answers
44 views

What is this graph called?

I'm not sure what it's called. Update: It's from the NED. Also explain: The name of these 4 axes. I mean I guess Y and X are in the same position but what about the others? The ...
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0answers
11 views

Best system ? Without martingale

I wonder what is the best system, if the percentage of winning bets is 52%. Bets have two options 1 or 2 and is rounded to the average odds 1.80, and the maximum wrong bets  in a  row is 8. So, with ...
2
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0answers
36 views

How to scale “probabilities” to a given mean?

I have a set of scores $x_i$, $i=1,\ldots,N$ (mimicking probabilities, $0\le x_i\le 1$) and I want to transform them so that the result has a given mean $m$, while remaining in the interval $[0;1]$. ...
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1answer
21 views

BINOMIAL PMF distribution [on hold]

X∼binomial(1,1/3) and Y∼binomial(2,1/2) Obtain PMF of W = XY + 1 How to approach and solve this problem. Can some one help me in this regard? Thanks,
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30 views

Can someone help me with the following math question/dilemma?

I have a pool of objects that are randomly selected from a global object database. The objects certain numeric attributes: The objects from the pool are fed to users in real time Users will either ...
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1answer
31 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
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0answers
17 views

Is logistic regression unbiased and efficient?

Seeing how in social sciences the Carmer-Rao lower bound is used as variances of the found parameters it would seem that the parameters are both unbiased and efficient, but what is the proof for this? ...
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11 views

normal dribution -with mean time and standard deviation [on hold]

The time taken by passengers arriving at an airport has been recorded as follows: The mean walk time from the chocks on of an aircraft that lands to the immigration counter is 5 mts, with a standard ...
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0answers
11 views

Creating trendline of one individual over the population data [on hold]

Lets assume I have collected the data about how often people give a coin to beggar depending on the weather outside. So in Y-axis there is donation% from 0% to 100% and X-axis represents how good the ...
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0answers
31 views

In how many ways can you make change for a dollar? [duplicate]

I know there are questions related to this on the site but they are not in the context I am looking for (basic statistics). This problem is at the end of the section introducing combinations and ...
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0answers
38 views

Rupam got a question.need help [on hold]

If theta be the angle between two regression line then how i will show that $\sin(\theta)$ less than equals to $1-r^2$? where r is correlation coefficient
3
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0answers
22 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...