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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Maximum likelihood estimate vs likelihood ratio tests?

Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a ...
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21 views

What is the expected value of this sum (drawing slips from a bag)?

Suppose there are thirteen slips in a bag, labelled 1-13. If I draw a 10, 11, 12, or 13 then I stop adding to the sum and return the sum. If not, I add the number I draw to the current sum and place ...
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89 views

Simplify this summation

$$\sum_{k=5}^\infty{{k-1}\choose{k-5}}\frac{k^3}{2^k}$$ I can't seem to simplify this sum. I get to a certain point then I get stuck, I know there must be some sort of trick to simplify it but I am ...
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2answers
9 views

calculating stdev from a previous stdev [on hold]

I am writing a program that tracks performance, is there a way to calculate stdev when the only data i have is the new value, old mean, old N, old stdev? eg. ...
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1answer
20 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
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1answer
44 views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
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10 views

Choice of the ridge parameter using leave-p-out estimation of the risk

The context of my question is the kernel ridge regression (in the field of Statistics). The question is to find a closed formula for $\lambda^{*}$ \begin{align*} \lambda^{*}=\underset{\lambda >O ...
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8 views

Sparse data and covariance matrix computation

Background I am trying to apply Gaussian Discriminant Analysis (GDA) on the MNIST dataset of hand-written digits, with 10 classes for 10 digits. In this dataset, each point is a vector of 784 ...
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1answer
22 views

Can't see how this equality that involves the gamma function holds

$$\frac{1}{\sqrt{2\pi}}\sigma^m \gamma\left(\frac{m+1}{2}\right) 2^{\frac{m+1}{2}} = 2^{\frac{m}{2}}\sigma^m \left(\frac{m-1}{2}\right) \left(\frac{m-3}{2}\right)\cdots \left(\frac{3}{2}\right) ...
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1answer
17 views

MSEP error derivation

I have random variable $X$ and set of observations $\mathcal{D}$. Let $\widehat{X}$ be an estimator for $E[X|\mathcal{D}]$ and predictor for $X$. Mean square error of prediction MSEP is defined as $$ ...
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1answer
27 views

Math formulas on Clustering

I am currently studying Clustering in Machine Learning. I have found a document regarding guessing the right number of clusters. I am reading the first part of it, having difficulties in understanding ...
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4answers
60 views

Min and Max of $ f(x,y)=\frac{x-y}{a-x-y}$

I'd like to find max and min of $$ f(x,y)=\frac{x-y}{a-x-y}$$ where $0\le x<y\le a/2$. Any one can suggest? Thank you
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1answer
36 views

Simple way to derive formula from dataset?

I'm not too good at math but I'm going to give this ago anyway and try and explain. Say I have a dataset, where each object in the set (not too sure about my correct use of terminology!) has $4$ ...
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1answer
30 views

Projection equation

I'm a programmer, not a math expert or statistician by any means, but my organization wants a page in our admin console that displays a projection of how many registrations we can expect to see based ...
2
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1answer
32 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
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1answer
24 views

Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
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1answer
15 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
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20 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
2
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1answer
25 views

What do I need in order to draw conclusions from this data?

I have three techniques, called A, B and C. Each can be used independently when trying to perform four related tasks (Tasks 1, 2, 3 and 4). I have run lots of tests, and tried all combinations of each ...
3
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1answer
33 views

Is it possible to determine the average and median if we only have the upper part of a bell curve?

Practical example: http://dota2.com/leaderboards shows the ~800 best solo ranked match-making scores in an Elo rating system. The lowest possible score is 1. There were 9318362 players last month, but ...
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8 views

variance-covariance matrix of 2sls in homoskedasticity and heteroskedasticity

How the variance-covariance matrix of two stage least square estimate in homoskedasticity or heteroskedasticity? The answer in homoskedasticity is $\sigma^2(E[xz'](E[zz'])^{-1}E[zx'])^{-1}$ and in ...
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21 views

Finding the underlying model from data

Let's say I have "k" data points in "n" dimensions. I also have "k" results of an unknown function of this data. Is it possible, simply given the data and the results from the function, to guess the ...
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13 views

L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
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1answer
21 views

Expected value of $(\overline{X} - 5)^2$?

$X_1, \ldots, X_9$ are 9 random samples from a N(5, 9). I am looking for the distribution ofExpected value of $(\overline{X} - 5)^2$. $$E[(\overline{X} - 5)^2] = E[\overline{X}^2 - 10\overline{X} + ...
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1answer
36 views

Do doctors have a shorter lifespan than the general population - Null Hypothesis

Do doctors have a shorter lifespan than general population with this data: Known lifespan for US subject in last 2 decades = 70.1, with a standard deviation of 7.6. A sample from 51 doctors from the ...
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2answers
61 views

Why is this expectation true?

Working with Rao-Blackwell, this came up: $$E[2X_1 \mid \max(X_i) = t] = 2\left(\frac{1}{n}t + \frac{n-1}{n}\frac t 2\right)$$ Where X are uniform(0, $\theta$). What are the intermediate steps? I'm ...
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1answer
24 views

Is there a significant difference in population percentages

The question is, "If an 85% confidence interval (CI) for the difference between the population of A Students (Pa) and B Students (Pb) is -15.8% < Pa-Pb < -7.9%, is there a significant ...
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14 views

Underperformance of an NCAA team

If we played a game where a NCAA team was worth 100 for winning, 50 for second place, 25 for QF, 10 for R16, 5 for R32 and 0 for R64, and we had our expected values are based on the oddskmakers ...
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3answers
133 views

Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
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Bias of the MLEs for the two-parameter Weibull distribution

Is it possible to obtain a formula for or an equation on the exact bias of the MLE-vector for the two-parameter Weibull distribution (both parameters unknown). I've read papers offering Monte Carlo ...
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1answer
46 views

Assigning $\pm 1$ values to the edges of a complete graph

I read this sentence in one combinatorics book. In graph $K_{100}$, there is a possible way to assigns number (value) from $\{+1,-1\}$ to each edge, so that the sum of all edge values connected to ...
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25 views

How to find mean and median from histogram

Solution for finding mean : The problem faced when i saw a video to evaluate the mean https://www.youtube.com/watch?v=vMrc6dP8pCo According to the video, the lecturer said that, we can take ...
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61 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
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2answers
34 views

If $X$ has CDF $F$, how can I find the CDF of $U= \max \{0,X \}$?

If $X$ has CDF $F$, how can I find the CDF of $U=\max\{0,X\}$? Obviously the suport of $U$ consists solely of nonnegative values. Am I right then in thinking that for $u=0, F_U (u)=F_X(0)$ and for ...
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1answer
21 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
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1answer
63 views

Combination Problem with Sofa [on hold]

Suppose we have 5 sofa on room A. in this room, 4 students seated on these sofa. These Strudents go to another room for eating dinner, and after that come back to room A. how many way the students can ...
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1answer
5 views

Interpretation of gross reproduction rate and net reproduction rate

If GRR=1.234 then what can be said about the population growth? And then what can be said when net reproduction rate, NRR=1.234?
2
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1answer
60 views

Sufficient statistic

Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently. The set ...
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1answer
20 views

How to compute the difference between two numbers using a weighted formula?

Currently at work we use the formula ((expected-actual)/expected)*100 to show the difference between two numbers, however, this formula shows extreme differences for small numbers. For example, in the ...
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3answers
108 views

In 30 boxes are 15 balls. Chance all balls in 10 or less boxes?

Question1: I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i found 0 balls. Afer i collected all 15 balls i put them randomly inside the boxes. How much is the chance that all balls are ...
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2answers
26 views

Adding statistical weight to reviews

I'm trying to prioritize a list of places using a few factors: Number of reviews for each place Percent of people that recommend each place (from the pool of people that reviewed) Overall rating of ...
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2answers
32 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
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1answer
67 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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2answers
36 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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3answers
71 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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0answers
15 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
27 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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For a normal distribution, find the percentage of data that are between 3.5 standard deviations below the mean and 3 standard deviations above [closed]

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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1answer
33 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
27 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, ...