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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Expected value over many trials

I am a serious 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 ...
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7 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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12 views

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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7 views

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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49 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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18 views

Asymoptotic distribution of identically distributed random variables [on hold]

$Y_1, Y_2, ..., Y_N$ 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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20 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
73 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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31 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
41 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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19 views

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 ...
2
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1answer
45 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
36 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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1answer
42 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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23 views

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
15 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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19 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
30 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

transform Normal CDF to Uniform? [on hold]

I am stuck in a problem below. This exercise (3.3) is from the book, "Introducing Monte Carlo Methods with R" "If we are interested in the tail probability Pr(X > 20) when X ~ N(0,1), simulating ...
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1answer
16 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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18 views

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
27 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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13 views

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 ...
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25 views

Statistics. Chapter 8

A random sample of 50 items result in a sample mean of 20. The population standard deviation is 10. What is the standard error of mean. At 96% confidence, what is the margin of error. Compute the 98% ...
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3answers
25 views

Problematic Permutation Problem

i see a problem without any definition. would you please help me? i want to calculate the number of permutations of 1,2,...,1392 that 696 numbers be in the natural positions (from all numbers, 696 ...
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1answer
27 views

How to simulate from a simple point process

Define a point process by the conditional intensity function $$\lambda^*(t) = \mu + \alpha \sum_{t_i < t} e^{-(t-t_i)}$$ where $\mu$ and $\alpha$ are positive parameters. I would like to ...
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40 views

Statistically determine whether a set of variables are diverging

I'm collecting data on a battery which is being discharged and charged again repeatedly. This battery consists of 4 cells in series and I am recording the fully-charged voltage of each cell at the end ...
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Find the mean and median help!

I feel like there isn't much information, but the answer for the mean is 79. The median is said to not be able to be found. Can someone explain this? Thank you!
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9 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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10 views

Difference between Kendall tau distance and Kendall tau rank correlation coefficient

What is the difference between Kendall tau distance and Kendall tau rank correlation coefficient? Which one I should use to calculate how similar two arbitrary permutations of ranks are? Suppose, we ...
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20 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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529 views

Closed form equation for win percentage of two battling armies

I was pondering a battle mechanic for a board game that is similar to, but simpler than battling armies in Risk. Consider one army of size X and a second army of size Y. The battle occurs by ...
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40 views

Probability Question involving Probability Mass Function/Random Variables

Problem: When a paging system transmits a message, the probability that it will be received correctly by the appropriate pager is p. To ensure that the message is correctly received at least once, the ...
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12 views

Produce a list of the most-similar units, given various correlations/relationships

I have a database full of units (U1 - U50, U51...) where every unit has the same standard attributes (A1 - A10) and where a % of each attribute defines the amount of that attribute for that particular ...
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1answer
27 views

Comparing the width or variance of two distributions with uncertainty on variance estimate

So I have two probability distributions ,one from data and one from a model. They are not Gaussian shaped but skewed. I have measured their rms or width by finding the median and the 16% (lower limit) ...
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32 views

Derivative of an exponentially weighted moving average

It has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however, ...
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12 views

Difference between ANOVA and Pearson Chi-squared test

To test the equal of variance the non categorical variable using ANOVA test. As for the data with categorical variables using Pearson's Chi-squared. In this case, why the ANOVA test can not be used ...
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1answer
11 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
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18 views

A simple plot of a list in R

This is a very easy question, I guess. But unfortunately I struggle to solve it in an easy way: I have a list in R. I would like to plot the entries of each column vertically and label the x-axis to ...
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17 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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19 views

Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
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1answer
32 views

Find a 10% likelihood interval

The function is: (n choose x)$[(1-y)^{k}]^{x}[1-(1-y)^{k}]^{n-x}$ Suppose n = 100, k = 10, x = 89 I found the maximum likelihood of y-hat to be 0.0116 Now I need to find a 10% likelihood interval. ...
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25 views

Two Reflecting Barriers

A chain with stats 1,2,....,n has a matrix whose first and last rows are (q,p,0,...,0) and (0,...,0,q,p). In all other rows Pk,k+1 = p, Pk,k-1 = q. Find the stationary distribution. I am ...
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How could you find the probability that the estimator is within 0.03 of the mean?

p = fraction of large population that smokes n = sample size y = # in sample that smoke The maximum likelihood estimate of p is p-hat = y/n Consider the random variable Y and estimator F = Y/n ...
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Simple question about showing independence

How does one show that it is possible for a random variable $Z$ to be independent of $A$ but also not independent of $X$ where $X=1\{A>B\}?$ Under what circumstances can this be true, ignoring the ...
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35 views

Conditioning on information about the moments of a random variable is trivial

Say we have some random variable, $X$. Is it always trivial to condition on information about the moments of $X$? For example, suppose we know that $\mathbb{E}(X)$ is positive. But ...