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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2answers
145 views

Compute the probability that $ | \bar{X} - \mu | > S$

Given $X_1, \ldots, X_n$ from $\mathcal{N} (\mu, \sigma^2)$. I have to compute the probability: $$P\left(|\bar{X} - \mu| > S\right)$$ where $\bar{X}$ is the sample mean and $S^2$ is the sample ...
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1answer
160 views

Measure-theoretic view of expectation of a Bernoulli sequence

Problem: I have a good understanding of basic Bernoulli and Binomial RVs, but this was foundational work in statistics. I am attempting to try and apply my (minimal but increasing) knowledge of ...
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1answer
33 views

For System dependent on normally distributed parameter, are deviations added or variations?

Say, A and B are two normally distributed parameters with their variations being $\sigma^2_a$ and $\sigma^2_b$. Now for system C, which is linearly dependent on these parameters, is its ...
2
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1answer
2k views

Distribution of the sample mean of a exponential

I please ask someone to check if my calculations are right. I have $X_1, ..., X_n$ from a $\mathcal{E}(\lambda): f(x, \lambda) = \lambda e^{-\lambda x}$. I have to find the $k$ such that $P(\bar{X} ...
0
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1answer
70 views

Variation Tolerance

I came across a statement in my course book that 3$\sigma$ is considered as a means of tolerance. Can anyone explain it to me. I understand that +3$\sigma$ to -3$\sigma$ constitutes 99% of the ...
1
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1answer
91 views

Touch Typing Index - Speed and Accuracy

I am trying to determine the ability of my students to touch type. I have data on their speed (in seconds) and their accuracy (number of errors). I also know the number of words in the test (50 ...
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0answers
447 views

Calculating a fisher information

$\ (X_1,X_2,X_3,X_4)$ has a multinomial distribution with parameters 3$\theta$/5, $\theta$/5,$\theta$/5,(1-$\theta$) Calculate the fisher information where $\theta$ is in [0,1]. So I wrote out the ...
0
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1answer
103 views

Linear regression for normal distributions

Basically, I have that $\ Y_i = \alpha +\beta(x_i-x_{bar}) + \epsilon_i $ where $\epsilon_i$ are i.i.d normally distributed with mean 0 variance $\sigma^2$ $\ Y_i ~~has ~a~normal~distribution~as ...
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1answer
412 views

Fitting of Closed Curve in the Polar Coordinate.

I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I'm thinking of deducing a ...
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0answers
34 views

Dynamic Light Scattering

In DLS does the combination of the Time Average and Ensemble Average give a better statistical average than the results shown by each case considered separately ?
4
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3answers
984 views

A question on confidence

So, I've been reviewing some of my old stats courses in preparation for an interview I have in a couple of days. I'm a bit stuck on a particular question and hope you could help. A drug trial gives ...
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0answers
63 views

calculation of the sum using idea of one answer

I am wondering if the sum (the $q$-th moment) in my question Calculation of the moments using Hypergeometric distribution can be calculated using idea in Evaluating 'combinatorial' sum ? ...
5
votes
1answer
613 views

Calculation of the moments using Hypergeometric distribution

Let vector $a\in 2n $ is such that first $l$ of its coordinates are $1$ and the rest are $0$ ($a=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2n\}$. Define ...
5
votes
5answers
640 views

What does it mean to do MLE with a continuous variable

I am struggling with the semantics of continuous random variables. For example, we do maximum likelihood estimation, in which we try to find the parameter $\theta$ which, for some observed data $D$, ...
2
votes
1answer
159 views

Is the expectation $E[\xi U'(\xi)]$ finite?

I encounter the following problem today. It seems a simple question. Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions: (1) $U$ is concave, continuous, ...
0
votes
1answer
410 views

discrete random variables transformation

I have two discrete random variables $X$ and $Y$, let $P_X$ and $P_Y$ be the PMF of the random variables, if $Y=X^2$ ,I want to know the PMF of Y in terms of PMF of X ? I know how to do it with ...
1
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2answers
1k views

How to transform normally distributed random sequence N(0,1) to uniformly distributed U(0,1)?

Everybody knows how to convert U(0,1) to N(0,1). However does anybody know an efficient algorithm solving the opposite task? I mean how to generate U(0,1) sequence from N(0,1) one? Asking because a ...
0
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1answer
132 views

Maximum Likelihood problem in book Pattern Recognition and Machine Learning

I ran into a problem in section 2.4.1 namely "Maximum likelihood and sufficient statistics" under "Exponential Distribution Family" of Bishop's Pattern Recognition and Machine Learning. excerpt How ...
3
votes
3answers
122 views

Detecting significant decreases in a signal

I'd like to find a way to detect a significant drop/decrease in a signal. Below is an actual example of what I'd like to accomplish, with the arrow denoting the change that I'd like to detect (only ...
4
votes
1answer
354 views

Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?

I am interested in finding a closed form solution (wich I suspect does not exist) to the following integral $$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x ...
1
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1answer
158 views

Tricky probability explanation

Question: How many times do we have to throw a normal coin so that, we are at least $99\%$ sure, that the percentage of heads will be between $49.5\%$ and $50.5\%$? Solution: $$\begin{align} ...
2
votes
1answer
580 views

“Flattening” a 2D Normal Distribution

I would like to model the probability of a point being at a certain place on a 2D grid. The X coordinate of the point varies according to a normal distribution with mean $0$ and standard deviation ...
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0answers
66 views

Stochasticity of Fermi problems

The great physicist Enrico Fermi was known for his ability to make good guesses with little or bad data by multiplying series of estimates. 1 I've seen this described as corresponding to a stochastic ...
4
votes
2answers
2k views

Sample variance derivation

I have quite a simple question but I can't for the life of me figure it out. For a set of iid samples $\,\,X_1, X_2, \ldots, X_n\,\,$ from distribution with mean $\,\mu$. If you are given the sample ...
2
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4answers
865 views

Simple question on standard deviation and mean.

This question has kinda stumped me: For a data set, the mean is 5 and the standard deviation is 1. There are 10 values in the data set. Is the range of the data set bigger, smaller or equal to ...
2
votes
2answers
279 views

An interesting problem about almost sure convergence

Assume that $X_1,X_2,\ldots$ are independent random variables (not necessarily of the same distribution). Assume that that $Var[X_n]>0$ for all $n$. Assume also that $$\sum_{n=0}^\infty ...
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1answer
223 views

Maximum Likelihood Example

I have the problem to understand the following simple Maximum (Log) Likelihood example. Let $X$ be a discrete variable with domain $\{1,\dots,K\}$ and the discrete distribution is parametrized ...
0
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1answer
211 views

Almost sure convergence of random variables

Assume that $X_n$ are independent (but not necessarily of the same distribution) and that $Var[X_n]>0$ for all $n$. We know that $$\frac{X_n-E[X_n]}{n}\to 0 \textrm{ almost surely as ...
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2answers
83 views

How to find a formula?

I'm trying to find out how some value is calculated in some strategy game. I know the variables that make that value, but not how they are combined to make it. So I gathered some samples: c=100 when ...
2
votes
1answer
71 views

local variance of Markov decision processes

Does anybody know the notion of "local variance" of Markov decision processes? Any reference would be appreciated. Thanks.
0
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1answer
91 views

Translate percantage increases / decreases into 0 - 100 range?

I am tracking some percentage increase and decrease numbers and I need to plot them on a line graph. However I can only plot them on a line graph with a range of 0 - 100. What formula would I use ...
2
votes
1answer
720 views

Constructing correlated random variables

How do I construct two correlated random variables with correlation $\rho$ given two i.i.d normal r.v.? Do I just multiply the correlation matrix by a vector generated with two i.i.d normal variables? ...
0
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1answer
58 views

What is a common way to measure the “goodness of fit” of an individual data point to a correlation?

Let's say I have a collection of data points (X & Y values) that show some correlation when, eg, Pearson's correlation formula is applied. What is a good measure for determining which data points ...
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vote
1answer
180 views

How long does it take to complete a sticker album?

We are collecing stickers in chocolate bars and whenever we open a bar we get a random new sticker. There are many different stickers and we try to collect them all. We open the first bar and get a ...
1
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1answer
2k views

A summation inside of the expected value operator

Here is part of the solution to my homework, which I could not wrap my head around. Given are $X_n$ random variables n = 1,2,...,N which are independent and Gaussian distributed with known mean $\mu$ ...
0
votes
1answer
104 views

Sampling a Gaussian in a Bayesian network

There is a 1-dimensional Gaussian random variable $x$ with $P(x) = \mathcal{N}(x\ |\ 0,1)$, where $0$ is the mean and $1$ is the variance. There is a binary random variable $Y$ with $P(Y = 1|x) = ...
3
votes
1answer
436 views

References for Kolmogorov's strong law of a large numbers

On the Wikipedia law of large numbers site, they mention "Kolmogorov's strong law of large numbers", which works even if the random variables are not identically distributed. Where can I find this ...
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2answers
202 views

Find the distribution of a transformation

Let $$f_X (x, \theta) = \frac{1}{\theta} x^{\frac{1}{\theta} - 1}, \; x \in (0, 1)$$ find the distribution of: $$Y = - \frac{1}{\theta} \ln X$$ [Solution provided: $Y \sim \mathcal{E}(1)$ ] I did: ...
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1answer
487 views

Unbiased estimators

A single observation is made from a poisson distribution with unknown mean $\lambda \geq 0$ However any value greater than 2 has been rounded down to 2. This we have the observed value of a single ...
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1answer
141 views

Ranking System and Separated Populations

I'm trying to modify the ELO ranking system formulas to adapt them to eSport (electronic sports, but more specifically Starcraft II). (The reason I'm using ELO, is the straight forward concept and ...
0
votes
1answer
104 views

What is the Jacobian and what is its importance relative to the constant for the multivariate distribution?

Familiar readers should recognize (13) as the exponential part of the multivariate probability distribution where $A$ is positive-definite. In trying to derive the constant for the distribution that ...
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1answer
151 views

Find cumulative, marginal densities, moments for $f(x)=2,\le y \le x \le 1$ (multivariate)

Im learning multivariate statistics on my own and I have come across some problems I don't understand. Unfortunately there's no solution manual in the back so I thought I might ask here. Its not a ...
2
votes
2answers
124 views

Normal distributions obey central limit theorem

Let $X_1,\dots,X_n$ be independent random variables, each normally distributed as $X_k\sim N(m_k;\sigma^2_k)$. Let $S_n = \sum_{k=1}^n X_k - m_k$ and $T_n = ...
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vote
2answers
313 views

Drawing three balls from an urn with ten balls

An urn contains four white balls and six black. Three balls are drawn with replacement. Let $x$ be the number of white balls. Calcaulate $E (x)$, $VAR(x)$ and $\sigma x$. I don't know how to ...
0
votes
1answer
91 views

When deriving the multivariate normal distribution, please explain this limit

In deriving first the bivariate case, I have come across this limit that confuses me, (7). It should be easy enough for a mathematically mature person to understand and explain. Is the limit a typo? ...
0
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1answer
727 views

Marginal PDF from a joint PDF with an integral that does not converge

I have a joint PDF that has gone through some transformations of $f(x,y) = 12x\displaystyle\frac{1-y}{y^3}$,$0<x<y^2$, $0<y<1$ It definitely is a valid PDF as it has a double ...
5
votes
1answer
223 views

$X$, $Y$ gaussian variables, $\mathbb{E}[X^2Y]$ and $\mathbb{E}[X^3Y]$ as a function of its means, variances and covariance?

Let be X and Y two not independent Gaussian random variables of means $\mu_X$, $\mu_Y$ and variances $\sigma_X$, $\sigma_Y$, respectively. Let also be $\Sigma$ the covariance between X and Y. I'd ...
3
votes
1answer
793 views

Examples of sufficient statistics for non-exponential family distributions?

I know that the Pitman-Koopman-Darmois theorem says that only exponential family distributions have sufficient statistics whose dimension stays constant as the sample size increases. I further know ...
0
votes
1answer
546 views

Math problem in Pattern Recognition and Machine Learning

I had been carefully following Bishop's Pattern Recognition book and came across some of difficulties due to my merely basic math ground.First begin by the Gaussian section on page 80. The ...
1
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1answer
392 views

expectation maximization in coin flipping problem - one more …

I am looking at the same coin-flipping problem - http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html?pagewanted=all I understand the example well, but am unable to correlate it with the ...