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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Finding the frequency distribution function of a given compound distribution.

A random variable $N$ takes the values 0 , 1 and 2 with probabilities 0.5 , 0.25 and 0.25 respectively. We let $S = \sum X_i$ where $X_i's$ are independent and identically distributed random variables ...
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1answer
37 views

Decomposition of Random Variable (Information)?

I am wondering whether the following idea or something similar appears in a field such as statistics or information theory(?). Take a random variable $Y$ which takes value $1$ or $2$ with equal ...
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1answer
56 views

1D random walk probability distribution

I am way more physicist than mathematician and this question arises from experimental physics/engineering. The motivation is dealing with small amount of random discrete shifts between measured ...
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8 views

RMSE for shifted series

I am facing the problem to compare a series of simulations to a set of observations. Normally I would employ the Root Mean Square Error (RMSE) as goodness-of-fit measure to understand which is the ...
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1answer
51 views

What at the chances of getting 20 heads on a row if tossed 100 million times? [duplicate]

I understand that each toss has a 50% chance if it is a fair coin, but I have hard time grasping the law of great numbers and I would like to know how likely it is that I get 20 heads in a row in such ...
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9 views

Combine sum of binomially weighted Poisson distributions

I have the sum $$ S(k,\lambda, n, p) = \sum_{m=0}^n P(k|m\lambda)B(m|n,p)$$ where $P(k|m\lambda)$ is a Poisson distribution with parameter $m\lambda$ and $B(m|n,p)$ is a binomial distribution with ...
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7 views

What is the relationship between regression line and graph of averages?

Why is the regression line regarded as a smoothed version of the line of averages, and why is it that when the graph of averages falls on a straight line, that line is also the regression line?
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1answer
24 views

the meaning and usuage of the variance of a discrete random variable

We know that the variance in continuous normal disturbed random variable, enable us to calculate confidence intervals. But I just have to wonder what is the point in calculating variance of a discrete ...
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1answer
32 views

Why is the regression line an estimate of the average value of y for each value of x?

The regression line, passing through the point of averages with a slope equivalent to r, is said to be a good estimate of the average value of y for each value of x. I can see why this is the cases ...
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45 views

How can I generate random samples from following probability density function?

Let $\mathbf{\alpha}=(\alpha_1, \ldots, \alpha_m)$. The posterior density function of $\mathbf{\alpha}$ is given by $$h_0(\mathbf{\alpha}|\mathbf{x})=‎\frac{\prod_{i=1}^{m}\alpha_i^{a_i}}{\left(1+\...
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2answers
87 views

Probability with biased coin problem

Jules César gives Astérix a biased coin which produces heads 70% of the time, and asks him to play one of the following games: Game A : Toss the biased coin 99 times. If there are more than 49 heads, ...
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22 views

Is it possible to test for non-normality (in a general way)?

I'm just curious and trying to get my head around this question. I feel like this requires applying a probability space structure to the space of all distributions that are not normal, but, I'm ...
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25 views

Expectation of the fill distance of $N$ random points in $[0,1]^s$

Let $x_1,\ldots,x_N$ be uniformly distributed points in $[0,1]^s, s \in \mathbb{N}$. What can be said about $$ \mathbb{E} \left(\sup_{x \in [0,1]^s} \min_{i \in \{1,..N\}} \|x - x_i\|_2 \right) $$ ...
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17 views

Applying delta method on euclidean distance

In order to estimate confidence interval of a k-dimension euclidean distance, I need to use delta method to estimate standard ...
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1answer
41 views

Probability Average amount of rolls

I have a question regarding probability. I'll start by saying I've never taken a statistics or other similar course and was trying to work out this for a game. On average how many attempts will it ...
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0answers
38 views

Is it possible to be a frequentist and a subjectivist at the same time?

I'm trying to understand the differences between (1) Bayesian vs frequentist; and (2) subjectivist vs objectivist. So far my understanding (correct me if I'm wrong) is that: (1) Bayesian vs ...
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1answer
37 views

Finding the overlap between direction of distance in position space and direction of distance in velocity space

There are two objects A and B that can be described in position space and velocity space. The position space describes the instantaneous positions of the objects while the velocity space describes ...
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1answer
71 views

Finding Size-Bias Distributions

For a RV $W$ with mean $\mu$, let $W^*$ denote the $W$-size biased distribution (so that $EG(W^*)=\frac{E(WG(W))}{\mu}$ for all functions $G$ for which the expectations exist). I would like to learn ...
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23 views

Statistical test for convergence in distribution

Please forgive my total ignorance on all things statistical. I have some sequence of distributions (actual data, a collection of real numbers for each positive integer, which don't appear to converge ...
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1answer
19 views

Best coefficient between two data sets

I want to determinate a sensor coefficient but I struggle with a basic math problem... Here are my values : ...
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1answer
31 views

Stats question bionomial distribution. [closed]

It is known that 47% of students at a large university are male. If we take a random sample of 200 students at the university, what is the approximate probability that less than half of them are male? ...
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1answer
25 views

How to find normal distribution that has a quadratic?

Let $X$ be a normal random variable with mean 1 and variance 4. Find $P(X^2 − 2X ≤ 8)$. (Answer key .86) My attempt $$P(X^2-2X\le 8)=P((X+2)(X-4)\le 0)$$ and this is where I am lost. I did the ...
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1answer
80 views

How to find a symmetric matrix that transforms one ellipsoid to another?

Given two origin-centered ellipsoids $E_0$ and $E_1$ in $\mathbb{R}^n$, I'd like to find an SPD (symmetric positive definite) transformation matrix $M$ that transforms $E_0$ into $E_1$. Let's say $...
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1answer
19 views

Alternatives to Kruskal-Wallis or one way Anova test for small size samples

I have a group of measurements 'grouped by year' and on each year I have only one recorded measure, such as the example below: ...
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4answers
870 views

How many unique ways are there to arrange the letters in the word HATTER?

How many unique ways are there to arrange the letters in the word HATTER? I can't wrap my head around the math to find the answer. I know that if they were all different letters the answer would be 6!...
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Does anyone know of a way to overlap multiple data effectively?

Like each of us has a separate group we belong to right? And everyone belongs to different groups. So how to know the number of people in each group and each group everyone belongs to. Like Venn ...
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1answer
67 views

Is $\mathbb{P}( X > Y) = \mathbb{P}( X+k > Y+k )$ true, where X and Y are random variables?

My intuition tells me that $\mathbb{P}(X+k > Y+k) = \mathbb{P}(X > Y)$ should be true, since there (should?) be a bijection between every single result between these two probability ...
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5answers
58 views

Combinations Proof [closed]

Show that $2^n = \sum_{i=0}^n \binom ni$. I tried expanding $\sum_{i=0}^n \binom ni$ into sum of: $\frac{n!}{(n-r)!r!}$, and found the common ratio and then used the sum formula, but I did not get to ...
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1answer
48 views

What does it mean for a pdf to have this property?

What does it mean for a probability density function $f(x)$ to have the following property? $$1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ I have tried a lot to ...
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2answers
39 views

When deciding the Hypothesis, do I, in some way, define the rejection region?

This has always troubled me a bit. When I choose my hypothesis, do I define in some way the rejection region [RR], or, do I do that by choosing the test statistic I want to use? By fixing the ...
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2answers
106 views

Confusing myself with probability

A plane uses $1$ hour from $A$ to $B$. It has $2$ motors. Both motors works with a probability of $0.995$. If one motor (or both) fails, the plane crashes. How big is the probability the plane can get ...
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1answer
27 views

Finding probability sample proportion is less than 33% assuming null hypothesis is true

Candidates 1,2 and 3 are running for a position in a company. Candidate 1 claims 38% favourability among all the voters. Assuming this is true, what is the probability that in a random sample of 500 ...
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1answer
23 views

Is there any bound for the joint probability when the conditional probabilities are difficult to calculate?

When the joint probability for events $A_1,A_2,A_3,A_4$ is desired, the chain rule is used as follows:$$ P(A_4, A_3, A_2, A_1) = \mathrm P(A_4 \mid A_3, A_2, A_1)\cdot \mathrm P(A_3 \mid A_2, A_1)\...
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1answer
30 views

what is the meaning of 'population' and 'sample' in statistics

In the context of finding the average weight of crows, the wikipedia article https://simple.wikipedia.org/wiki/Population_%28statistics%29 mentions that the 'population' is the set of measurements of ...
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10 views

Summarizing set of parameters with different weights

I've an educational game which teaches several competencies sorted on relative difficulty in the ascending order. For each competency, I create challenges in the game, and during runtime following ...
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2answers
36 views

Need help understand independent identically distributed random variable.

Let $X_1, X_2, X_3$ be three independent, identically distributed random variables each with density function $$f(x)= \begin{cases} 3x^2 & 0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases} $$ ...
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20 views

Need help with cumulative distribution function problem.

A commercial water distributor supplies an office with gallons of water once a week. Suppose that the weekly supplies in tens of gallons is a random variable with pdf $$f(x)= \begin{cases} 5(1-...
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25 views

Softmax Derivation Help

I've been reading a paper that derives logistic regression from a few assumptions . Here is the link. If you go to page 5 and look at equation 18 the author claims that this essentially says the ...
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1answer
97 views

What is the average distance of two points chosen uniformly on a unit square?

What is the average distance of two points chosen uniformly on a unit square? What I am asking is how to calculate $E\left(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\right)$ for $x_1, x_2, y_1, y_2$ spread ...
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0answers
45 views

Is my SE ranking statistically unlikely? [closed]

I recently noticed --not that I was checking! --that my ranking on the Philosophy Stack Exchange is currently #8 for all time, year, quarter, month and week. It seems statistically unlikely that ...
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1answer
24 views

From a set of numbers, odds of a number being/not being selected when drawn randomly x times

Apologies if the title is unclear, This is my first question here. Anyway, Say I have a set of numbered balls i.e (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). I want to make a 15 digit number by randomly drawing ...
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1answer
19 views

Calculating variance for a window of samples which already contains pre-calculated variances

In a previous answer, the following solution was given for calculating the variance from a stream of sample values (from Knuth via John D. Cook): $$ \begin{align*} m_k&=m_{k-1}+\frac{x_k-m_{k-1}}...
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0answers
27 views

Exercises with solutions for mathematical statistics

I'm currently studying the statistics part of the book Georgii: Stochastics, contents are here (chapters 7 - 12). Sadly, there are no solutions for the exercises given in this book. Do you know a ...
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21 views

Build skew normal distribution knowing the mean, max, and min

Say I have a data point with included errors and I want to build some continuous distribution around it. Normally this might be a Gaussian because one knows the sigma and mean right off the bat. ...
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31 views

box and whiskers plot

I am having some troubles understanding what the components of a box and whiskers plot mean. Let's say I have some observations. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ...
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1answer
71 views

explain why accounting data follows Benford's law

I've spent a lot of time recently investigating on Benford's law. Many sources state that Benford's law can be useful when auditing accounting data. Non-fraudulent accounting data (under "good" terms) ...
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1answer
20 views

sufficient statistics to estimate the unknown parameters

I am a beginner in statistical inference and am learning sufficient statistics. As far as I know the distributions conditional on the sufficient statistics doesn't depend on the unknown parameters. I ...
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1answer
40 views

Conditional probability question with boxes and marbles - confused

There are n + 1 boxes labeled from 0,1,2,...,n. In each box, there are n + 1 marbles. Box 0 has 0 black and n + 1 white marbles. Box 1 has 1 black and n white marbles. In general, Box i has i black ...
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1answer
28 views

is the probability of having a specific pair of normally distributed variables depends only on the distance of their standard deviation from the mean

We have 2 independent random variables $a \sim N(0,1)$ and $b \sim N(0,1)$. I want a proof (or even good intuition) to show that the chance of having the pair $(a_0,b_0)$ is a function of only the ...