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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What to call & how to compute errors in a very asymmetric sample

Consider the following sample $\{1.25,1.5,1.75,2.0,2.25,2.5,2.75,10.\}$. Mean is $\mu=3.$, standard deviation is $\sigma\approx 2.69$. I am wondering how to compute and what to call error bars in the ...
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9 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
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1answer
15 views

Can any function of the second moment of a random variable be recovered from its quantile function?

Summary of question It is known that the expected value of a random variable can be obtained from integrating its survival function. This is easily restated in terms of the quantile function as: $$ ...
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1answer
34 views

I don't understand how I can calculate this

"Let $X_1, X_2, ...$ independent random variables. $X_n\sim B(p_n)$ and $p_n = \frac{1}{n}$. Calculate $P(lim sup (X_n=0, X_ {n+1}=1,X_ {n+2}=0 ))$" I suppose that i can use the lemma of ...
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1answer
20 views

Formula to represent 'equality'

I am trying to find the appropriate formula in order to calculate and represent the 'equality' between a set of values. Let me explain with an example: Imagine 3 people speaking on an a TV Show that ...
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3answers
55 views

Why do we calculate the median the way that we do?

If we have 4 Apples: A A A A Then in any stats class, the medium is 2.5. I am very sure that this is wrong, and that the medium is actually 2, even though experts in the field would say otherwise. ...
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1answer
21 views

What's the relationship between half of a value x, and the midpoint of that value?

Say you have 4 apples: A A A A Where is the middle of this, I mean, on paper it's obvious, but what is the numerical value of this? Halving this value (or rather any value) is easy, 4/2 = 2, but is ...
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16 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
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28 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
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1answer
13 views

chi-square test as likelihood-ratio test

$X_1, \ldots, X_n$ iid $\sim N(0, \sigma^2)$. I want to deduce the chi-square test $H_0 : \sigma^2 = \sigma_0^2, ~ H_1 : \sigma^2 \neq \sigma_0^2$: Decline $H_0$, if $ \displaystyle \frac{(n-1) ...
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3answers
93 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
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16 views

finding a question about constrained regression - had a side constraint $x \geq y^*z$

I saw a question that asked about solving a non-negative least squares problem with $3$ unknowns, $(x,y,z)$. But there was an additional constraint, $x \geq y^*z$. Would appreciate getting the ...
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0answers
22 views

removing a factor when the summation equals zero

http://www2.warwick.ac.uk/fac/soc/economics/staff/vetroeger/teaching/po906_week567.pdf On page 9 of the above link. I don't understand the step from row 5 to row 6. After the x_i is pulled out as a ...
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0answers
60 views

Why is this the solution?

I have this exercise with the solution. Let $X\sim N(0,1)$. Show that $P(2X = 3Y + 1) = 0$ if $Y\sim \text{Poisson}(\lambda)$. I have this solution $P(2X=3Y+1)= P(\bigcup_{k=0}^{\infty}(2X = 3Y+1, ...
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0answers
18 views

Linear Combination of the Normal Distribution Two missing variables? [on hold]

An insurance office buys paper by the ream, 500 sheets, for use in the copier, fax, and printer. Each ream lasts an average of 4 days, with standard deviation 1 day. The distribution is normal, ...
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1answer
18 views

What is meant by statisticians when they talk about between population differences vs within population differences?

Suppose we have two populations of people in different parts of the world and we want to talk about the variation in heights between the two populations. As I understand it, statisticians are talking ...
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21 views

Median of the mean/max of an iid sample of exponential variables

I would like to know if one can obtain a simple analytic expression of the median of $T_n$ defined by $$ T_n = \frac{\overline X_n}{X_{(n)}}, $$ where $\overline X_n$ is the empirical mean and ...
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1answer
31 views

Expected value of discrete functions. [on hold]

I am doing some research in number theory(High school-so nothing advanced). During this I came across this post. I have not done much statistics. So could someone explain to me why if $\displaystyle ...
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2answers
34 views

Rayleigh Distribution: MLE biased?

This is most of an exam question I am doing for revision- some parts I have completed, others I am not sure about. We have $H$ the maximum height(depth?) of a river each year, modelled as a rayleigh ...
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1answer
35 views

probability of a “boat” in texas holdem [on hold]

If I have a pocket pair, what is the probability that I will receive a "boat" on the flop. A boat being getting one card of my pocket pair denomination, and 2 cards of another denomination?
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1answer
63 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
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14 views

How to analyse data samples scattered in time?

I want to analyse data corresponding to events happening at arbitrary moments in time, and conveying quantitative information. My goal is to study the relationship between the sum of these quantities ...
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6 views

Statistical Significance of Modified Data

I would like to know the statistical significance of achieving 1 or 7 100% of the time in a dataset of nine possible values. The nine possible values are the digital roots of a string. In a random ...
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1answer
41 views

Are there simple unsolved problems in statistics?

In number theory, calculus and various fields of mathematics, there are many unsolved problems. But are there simple unsolved problems in statistics?
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19 views

(x/y) v (y/x) as predictor in regression

I am trying to predict a variable 'score' with x and y. I believe it is related to (x/y)/z and z/(x/y), but I'm not sure which. Here is some data: ...
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2answers
34 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
4
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1answer
34 views

Same Expected Value but different variances. Is $E[U(X)] \ge E[U(Y)]$?

Let $U: \mathbb R -> \mathbb R$ be a concave function, and let $X$ be a random variable with a normal distribution, expected value $\mu$, and standard deviation $\sigma$. Let $\lambda \gt 1$, and ...
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1answer
35 views

How to Calculate Probability in class math? [on hold]

Suppose that $22\%$ of MATH students get an A grade, and $67\%$ of MATH students are from Berlin. The probability of being from Berlin or getting grade A is $75\%$. $S$ is from Berlin. Find ...
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0answers
16 views

Question - Opinion poll [on hold]

I've done 1 and 2, but struggling to do 3 & 4: It's about an opinion poll for opening a new store. - There can only be two outcomes ('yes' or 'no') - Respondents were a random sample $n=400$ from ...
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1answer
23 views
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1answer
40 views

Prove that $E[U(X)] \ge E[U(Z)]$

Let U: $\mathbb R$ -> $\mathbb R$ be a concave function, let X be a random variable with a finite expected value, and let Y be a random variable that is independent of X and has an expected value 0. ...
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1answer
23 views

construct confidence interval from proportions

Suppose you have a population of count data, i.e., $1,2,3, \dots, k$, you have a sample of the population of size $n$, and you have a confidence interval for the proportion of $1$'s , $2$'s,\dots$n$'s ...
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2answers
38 views

Confidence interval for Poisson distribution coefficient

This is an exam question, testing if water is bad - that is if a sample has more than 2000 E.coli in 100ml. We have taken $n$ samples denoted $X_i$, and model the samples as a Poisson distribution ...
1
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1answer
42 views

Generating a rating between 1 to 5 accounting for number of users who have rated

I need a rating system for my app, what happens is that a user can rate a thread 1 to 5. The calculation I was going to use was shown below. $$\begin{array}{llll} ...
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1answer
29 views

The asymptotic equivalence of LR, Wald and score tests

Suppose that $Y_1, \ldots, Y_{n}$ are iid from a Bernoulli distribution with parameter $p$ and consider $H_0 : p = p_0\,.$ The test statistics are $$ T_W = \frac{n ({\widehat p} - p_0)^2}{{\widehat ...
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2answers
31 views

Is standard deviation sensitive to arithmetic mean?

I have two sets of data, A and B, their arithmetic means are $\bar{A}$ and $\bar{B}$, and standard deviations are $\sigma_a$, $\sigma_b$, if A and B has the same sample size, and $\sigma_a < ...
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0answers
28 views

Question: Opinion poll [on hold]

I've done 1 and 2, but struggling to do 3 & 4: It's about an opinion poll for opening a new store. - There can only be two outcomes ('yes' or 'no') - Respondents were a random sample $n=400$ ...
2
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4answers
83 views

What does this definition mean: $F_Y(y) =P(Y<y)$?

I am doing calculations on $F_Y(y) := P(Y<y)$, but I am clueless as to what $P(Y<y)$ means. For instance the following question: Given function: $f_X(x)= 2\lambda x e^{-\lambda x^2}$ when $x ...
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2answers
67 views

Non-linear regression fit

I'm trying to fit my data to the following equation: $$ Y = A(1-2e^{bx}) $$ What I tried to do was transform the equation to a linear form via the following steps: \begin{align*} & A-Y = ...
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0answers
35 views

What does $x^{-}$ mean?

When I was reading "Probability and Statistics" (DeGroot & Schervish, 4th ed, p.110), I came across this equation: $$\Pr (X < x) = F(x^{-})$$ What does $x^{-}$ mean? I've seen $x^{10}$ or ...
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1answer
16 views

How to make a ranking of teams for their consistency in performance?

Question from my assignment: Collect the data of performance of all teams played in Indian Premier League (IPL) 2014. Using coefficient of variation check the variability in performance of ...
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2answers
54 views

Probability of selecting same numbers by players (not a pigeon hole problem)

There are $n$ players who play a game of selecting numbers from a range $k$ over a period of time and the rules of the game are as follows : The game is played for 1 minute and the players are only ...
0
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1answer
13 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
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22 views

Tying together the concept of the gamma distribution and its use in chi-square distributions

My understanding of the gamma distribution is with its association to poisson processes, where the gamma basically describes the probability that a given event number will occur in a specified time ...
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10 views

Hypothesis testing for $H_0 : \lambda \notin (\lambda_1, \lambda_2),~ H_1: \lambda \in (\lambda_1,\lambda_2)$

I want to test the hypotheses $H_0 : \lambda \notin (50, 55),~ H_1: \lambda \in (50,55)$ for the sample $X$: 30 24 68 63 59 53 where $X \sim Po(\lambda)$ and level $\alpha = 0.1$. I know that the ...
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0answers
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What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
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14 views

Quantile function of the sum of two normal random variables

I'm trying to proof linear risk aggregation. Assume $X $ follows $N(\mu_x,\sigma_x)$, Y follows $N(\mu_y,\sigma_y)$, and $S=X+Y$. In linear risk aggregation, they claim $F^{-1}_S = \sqrt{ (F^{-1}_X)^2 ...
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1answer
35 views

Statistics- Finding Probability

A local lawn service has determined the average time it takes to mow an average residential yard is thirty-five minutes. If mowing times are independent and constant, what is the probability it will ...
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1answer
27 views

Central limit theorem - generalizations [on hold]

I am looking for some generalizations for the Central limit theorem in the following sense: Let $\phi$ be a function on the natural numbers, under what conditions on $\phi$ $ ...
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1answer
30 views

what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...