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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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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19 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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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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39 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
21 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
20 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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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
30 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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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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19 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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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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3answers
42 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
19 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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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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20 views

Numerical example of SVM [closed]

Note. This question has also been asked on stats.SE. I have a constant problem understanding SVM for both linear and non-linear separable cases. I understand up to a point that SVM establishes a ...
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1answer
35 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
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
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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1answer
39 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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33 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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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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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
37 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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2answers
54 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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1answer
70 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 ...
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1answer
30 views

Finding the range from standard deviation and Gaussian Curve

The figure above shows a normal distribution with mean m and standard deviation d, including approximate percents of the distribution corresponding to the six regions shown. Suppose the heights of a ...
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Is it plausible that multiple rankings can be averaged?

Say you have multiple advisors, each of which may have multiple domains of expertise. You rank the advisors in each domain, starting at $1$ and proceeding through $2, 3,$ etc. Since an advisor can ...
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why not applicable categorical variables using ANOVA F or Levene test?

For independent test data is a categorical variable using the Pearson chi-squared test. But, why not applicable categorical variables using ANOVA F or Levene test? Are there any theories on this ...
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2answers
60 views

One of the Actuary Exam P question.

For Company A there is 60% chance that no claim is made during the coming year. If one or more claim are made, the goal claim amount is normally distributed with mean 10,000 and standard deviation ...
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1answer
23 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
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1answer
21 views

Histogram/binning data

According to a task we are supposed to bin data as follows: (1) 2001 (?) bins for the range from -1 to 1 in steps of 0.01 (2) 33 bins for the range 0 to 160 in steps of 5 I am not sure though, if ...
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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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63 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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2answers
43 views

can I get the correct average of a set of numbers from the averages of several subsets?

Let's say I have this set of numbers: 565 212 812 895 443 73 468 900 299 993 252 740 291 112 (average 503.9285714286) I'd like to split them apart into 3 sets ...
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16 views

Transforming uncertainties in spherical coordinates

I'm not looking for a very rigorous answer, it's a practical problem that I've encountered and whichever solution is best will be determined by trying it out on my data. I have a measurement of an ...
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34 views

Simple Variance approximation I don't get

I have $$ \log(\lambda_j -1) = c + \alpha_{j-1} $$ From here I know $$ \lambda_j -1 = \exp(c+\alpha_{j-1}) $$ Then, they say $$ Var(\hat \lambda_j)\approx\ Var(\hat \lambda_j - 1) \approx ...
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What is sampling with unequal probabilities without replacement

sampling without replacement: i know if my sample size is $n$, and if $i^{th}$ unit is drawn then we will not replace the unit in the population , so that in my next draw i can get any unit except ...
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1answer
46 views

I need help understanding this proof about convergence in distribution

The proof says that we used the fact that $(1-\epsilon)^\frac{x}{\epsilon} \rightarrow e^{-x}$ Why is this so? How do I prove this? Also, why do we need the fact that $\lfloor x/p_n \rfloor - ...
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2answers
37 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
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48 views

Upper Bound on Mutual Information

I am interested in an upper bound on mutual information that I have been encountering frequently in the statistics and probability literature. I have yet to see the "purest" form of the inequality, so ...
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Hypothesis testing if two random variables come from a related underling function. [migrated]

I have observations of two random variables $S_1(t)$ and $S_2(t)$ and I'm trying to build a test to see if they are related. In other words I'm trying to test if unknown functions $F_1(t)$ and ...
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1answer
17 views

Deriving marginal effects in multinomial logit model

For the multinomial logit model, it holds that: $$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$. Now my book states that the marginal effect is as ...
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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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1answer
32 views

The definition of the sample standard deviation

I am reviewing the statistic. I read the book "Probability and Statistical Inference" which is written by Robert V. Hogg and Ellit A. Tanis. There is a statement in the book says that: (Section: ...
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1answer
18 views

Features of a Set Given Same Value for Maximum and Average

What else, if anything, can be said if the maximum of a set is equal to the average value in the set? If the average is equal to both the maximum and minimum then all the values are the same, but can ...
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1answer
11 views

Stats “AND” Elimination

What's the logical behind eliminating this AND in a solution? p(he lives to 80 GIVEN he lives to 50) = p(he lives to 80 AND he lives to 50)/p(he lives to 50) = p(he lives to 80)/p(he lives to 50) ...
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Definition of Time Series

Having not done any stats for a few years, I seek clarification regarding the definition of time series given in my textbook. I apologize for the length, but I would be glad to just resolve my main ...
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1answer
30 views

What can we say about correlation coefficients?

If we are looking at sales and inventory data with a correlation of 50%, what can I conclude? My intuition tells me that 50 % of the movement in one of the variables is attributed to the others ...
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1answer
25 views

generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
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Correlation Statistic with a neutral score

I'm trying to develop a correlation statistic to compare lines/random variables (represented as a series of points). I want the correlation value to convey little information if one the lines is a ...