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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Help with probabilities on a game I am making

I asked over at the RPG stack exchange and they sent me here. I am working on making a RPG and am trying to understand the statistics of the core mechanic so I can determine how effective leveling ...
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1answer
103 views

What is the $P( |X-10| > 2)$ of a normal distribution when mean is 10, and standard deviation is 6?

I couldn't figure out this question: What is the $P( |X-10| > 2)$ of a normal distribution when mean is 10, and standard deviation is 6?
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27 views

Basic expression mangling question

I am looking at the solution of a physics/statistics hybrid exercise, and I can't figure out how one expression step took place. I have that $${dx \over dt} = gt$$ $$T = \sqrt{2h \over g}$$ where ...
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1answer
140 views

How to do a statistically sound ranking using varying numbers of likes/dislikes?

Assume the following problem: A popular website allows for users to either like or dislike its listed items (may it be movies, goods, people, or whatever). Each user may cast up to one like/dislike on ...
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64 views

Decide probabilistically whether leaf labels in a decision tree sum to zero?

I have the following problem, which might or might not be very easy to answer for someone with even a light background in statistics - but I don't even know where to start. Hence, I will give it a ...
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105 views

Probability of collision (two bivariate normal distributions)

I am trying to solve this problem on and off for the past couple of months but to no success. This was supposed to be a very small part of my PhD thesis in navigation but I guess I underestimated the ...
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75 views

Maximising the correlation coefficient

How do I show that the correlation coefficient $\rho(Y,f(X))$, where $f$ is any measurable function, is maximized in absolute value when $f(X)$ is linear in $E[Y|X]$. I know that for 2 r.v.s X and Y ...
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564 views

Are squares of independent random variables independent?

If X and Y are independent random variables both with the same mean (0) and variance, how about $X^2$ and $Y^2$? I tried calculating E($X^2Y^2$)-E($X^2$)E($Y^2$) but haven't been able to get anywhere. ...
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54 views

Parameter optimization in probabilistic models

Task: Suppose we model a variable $y = Wx + \mu$ as a linear transformation of $x$ plus some Gaussian noise $\mu\sim\mathcal N(0,\sigma I)$. Our aim is to minimize the estimation error of $x$ given ...
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94 views

Fibonacci like monkey population problem ( population dynamics )

Assume a female and male monkey. Lets call this generation $1$. All males and females have one and one only partner for breeding in their entire life assuming there is at least $1$ available. And ...
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668 views

to find the probability of a biased coin?

my ques is a bit different one, i don't want to know the probability of throwing coin n times or knowing the probability of biased flipping x times. i want to know the probability p of head, if a coin ...
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1answer
77 views

Multiplying by centering matrices

Suppose we have $X_{p\times n} \sim N_p(\mu, \Delta \otimes \Sigma)$. Then what is the distribution of $H_p X H_n$ where $H_p = I_p - 1_p1_p^T/p$ and $H_n = I_n - 1_n 1_n^T/n$? I know that it should ...
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1answer
173 views

How is Compound Interest Calculated between 2 Time Intervals?

I am building an application, where I need to understand how Compound Interest is calculated. I know, how to use the formula if the time is say 1 month, 3 months. If the duration is 42 days, How do ...
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2answers
611 views

How to approach probability questions?

I have four questions: The time that it takes to assemble a piece of machinery is well modeled by the normal distribution with mean of 72.9 minutes and standard deviation of 8.55 minutes. What is ...
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1answer
164 views

Some Questions on Determinants and Geometry

For real valued matrices, I know that the absolute value of the determinant is equivalent to the volume of the vectors forming the parallelepiped in the matrix. Suppose that $A$ and $B$ are real ...
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299 views

Balls in a box probabilities

A box is filled out by $1,000$ balls. The box can be thought of as containing $V$ sites and $V$ balls, with $V=1,000$. The box is repeatedly shaken, so that each ball has enough time to visit all ...
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400 views

Probability - Snow White and the 7 Dwarves

I'm stuck and I really need some direction as to how to tackle this problem. Each morning, before they go off to work in the mines, the seven dwarves line up and Snow White kisses each dwarf on the ...
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1answer
437 views

Can a chi squared distribution with a huge number of degrees of freedom be computed with a good precision?

Let $X$ be a chi squared variable with $121$ degrees of freedom. So the density $f_X$ of $X$ is defined by $$ ...
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1answer
46 views

How do I know an outlier in a time series isn't the beginning of a trend?

I wish to average detections coming in over time. I use the interquartile range to identify outliers and to discard them. I look at the last 30 values. What do I do if each new value is an outlier, ...
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666 views

Are outliers possible with categorical data

Just want to make sure that I understand the meaning of an outlier. Question: Can you have an outlier of categorical data? I think that to have an outlier you must first have some sort of ...
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1answer
2k views

Average proportion for proportions with different denominators

Say I have an experiment in which subjects are asked to respond to some stimulus. Their responses are transcribed and coded, first as "Valid/Invalid", and then for the valid responses, ...
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76 views

How to prove that estimator is plug-in estimator?

I need to prove that: a) sample median is a plug-in estimator b) sample quantile is a plug-in estimator, but I have no idea where to start; In a) : $$M = \begin{cases} ...
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1answer
109 views

Measure the uniformity of distribution of points in a 2D square

I am currently running into this problem: I have a 2D square, and have a set of points inside it, say, 1000 points. I need a way to see if the distribution of points inside the square are spread out ...
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35 views

How do I determine Heavy Tails on an empirical distribution?

How do you determine if an empirical distribution has a heavy tail? What would I have to do in order to determine that? I'm currently using mathematica, so if you know of any coding, that would be ...
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2answers
1k views

Statistic Missing Value

A professor has recorded exam grades for 30 students in his class, one of the 30 grades is unavailable. The mean score on the exam was 82, and the mean score of the 29 available scores is 84, What is ...
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4answers
1k views

Where does the guassian function/normal or bell curve come from?

I am confused as to where the function for the normal distribtuion comes from. Where does the e and pi come from? In my textbook I am presented with the function,but I am unsure about where it came ...
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1k views

Stratified random sampling

Which of the following is NOT a characteristic of stratified random sampling? (A) Random sampling is part of the sampling procedure. (B) The population is divided into groups of units that are ...
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4answers
258 views

Benford's law with random integers

I tried testing random integers for compliance with Benford's law, which they are apparently supposed to do. However, when I try doing this with Python, map(lambda x:str(x)[0], [random.randint(0, ...
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2answers
3k views

Normal Distribution, The “Y” Value

Guys I am having trouble with the standard normal distribution. http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm We know the X values run from approx $-\infty$ to $+\infty$ but ...
2
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1answer
391 views

Formulae for combining statistical moments

I am writing code to calculate statistical moments (mean, variance, skewness, kurtosis) for large samples of data and have the requirement of needing to be able to calculate moments for subsections of ...
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1answer
268 views

Prove that the Cartesian coordinates of a Rayleigh distribution are independent

I'm stuck on a question with a weird twist on the usual Rayleigh distribution. Instead of assuming that the components of the distribution are independent, we are given alternative conditions and ...
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1answer
84 views

Question about variance and its relation to standard deviation

I understand from my lecturer that variance an standard deviation are central to statistics. I do not understand the signifigance of both values, except that both measures the variability, and ...
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4k views

Expected value of a normally distributed random variable

How do you compute the expected value of a random variable? The problem I found asks; $$ W = rV^3$$ where $r$ is a constant and $V$ is a normally distributed random variable with mean 6 and standard ...
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110 views

Covariance matrix of a set of spherical coordinates

How do I compute the covariance matrix of a set of spherical coordinates? I know that I can compute the mean of a set of spherical coordinates by transforming them to cartesian coordinates, compute ...
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1answer
662 views

bounding the the expected value of the maximum of two random variables

Consider two standardized random variables $x$ and $y$, and define a function $g(x,y)=E[max(x,y)]$ where $E$ is the expected value operator. My question is finding the upper and lower bounds of ...
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3answers
680 views

1/f “Pink Noise” for the Math-Disabled

I have very little skill when it comes to math beyond all the elementary level operations (addition, subtraction, multiplication, division, mean, mode, etc) and a vague grasp of statistics, what a ...
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2answers
148 views

Calculating the density of a joint distribution

For the joint density function $$P\big((X,Y) \in A\big) = \int_A f_{(X,Y)} (x,y) \, dx\,dy$$ how would you show that if $(X,Y)$ is a random vector in $\mathbb{R}^2$ with density $f_{(X,Y)}$ and ...
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1answer
121 views

Why is the greatest and least values for each group in these 2 frequency distribution unknown?

Question: Write a few sentences comparing the distributions of P-T ratios for states in the two groups (west and east) during the 2001–2002 school year. Sample solution: The shapes of the two ...
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2answers
104 views

Determining most likely Gaussian distribution

I have two Gaussian distributions with mean and variance $(\mu_1,\sigma^2_1$) and $(\mu_2,\sigma^2_2)$. I then receive a series of values $x_1, x_2,...,x_n$ with mean and variance ...
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2answers
3k views

Casio fx 991ms Factorial

I apologize if this is the wrong forum for this, but I can't find the answer through google. I'm wondering how to find factorials on my casio f-991MS calculator. Any help is much appreciated.
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759 views

Variance of Ratio of Two Random Variables

Suppose we have two random variables $X$ and $Y$ with means $\mu_x, \mu_y$ and variances $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$. How would we derive $\text{Var} \left(\frac{X}{Y} \right)$? Edit. $X$ ...
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64 views

How do I show expectations according to this distribution?

Let $A$ (or $X$) be $\log A \sim N(\mu,\sigma^2)$, (lognormal distribution) I have to show $$E[A] = \exp[\mu + (\sigma^2/2)]\mbox{ and }E[A^2] = \exp[2\mu + 2\sigma^2].$$ Do I have to use mgf ...
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1answer
1k views

Calculating Variance of a binomial distribution using the standard formula $E(X^2) - \mu^2$

Binomial problems: Mean and standard deviation Suppose that the New England Colonials baseball team is equally likely to win any particular game as not to win it. Suppose also that we choose a random ...
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2answers
156 views

How is it that the required sample size for a specified error and confidence is not dependent on population size?

When calculating confidence intervals for population parameters, the population size is never a factor, rather sample size and the estimated parameter are used. It seems to me very counter-intuitive ...
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4answers
251 views

What gives rise to the normal distribution?

I'd like to know if anyone has a generally friendly explanation of why the normal distribution is an attractor of so many observed behaviors in their eventuality. I have a degree in math if you want ...
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1answer
77 views

Simple Linear Regression

Show that the estimated regression equation can be written as: $\hat{y} = \bar{y} + \hatβ_1(x - \bar{x})$ Say what this tells us about the fitted regression line. Okay so I know $y_i = β_0 + ...
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215 views

rate of convergence of mean ,variance & skewness estimators

We are asked a question which of mean,variance or skewness converges faster. At first I thought it was straight forward answer: mean->variance->skewness. But I am not sure anymore because I read ...
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47 views

Random Sampling around a Random Sample

Imagine I want to randomly sample how fast a driver is going on a highway. And for this purpose, let's assume that the distribution is normal with the mean as the speed limit. Now if I take a sample, ...
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1answer
275 views

Calculating P(A>B), where A and B are normal distribution

In the problem we have that $A \sim N(7, 11/60)$ and $B \sim N(7.3, 7/20)$ and the question is what is the probability that $A$ gives a higher value that $B.$ Since the textbook we have for the course ...
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260 views

Markov's Theorem

I'm having a hard time understanding Markov's Theorem. Maybe because I'm very tired, because I think this is a very easy question, but I can't wrap my head around it. For any set of nonnegative data ...