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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Why does this covariance matrix have additional symmetry along the anti-diagonals?

In my self study of a statistics book, I came across a page that has confused me somewhat. I am already familiar with covariance matricies, (or maybe not!), and the author's explanation leaves me a ...
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66 views

evaluating expected values or integrals

When $x$ and $y$ are two iid random variables, I want show if $$E(x\mid x>y)<1$$ can be determined: [1] without knowledge of the distribution of the random variables; and [2] if $x$ and $y$ ...
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91 views

Nonparametric method - Parzen windows

I don't understand the main function of the Parzen Window Let $u=[u_1, u_2,..., u_d]$ and define a window function $φ(u)=\left\{ \begin{array}{l l} 1 & \quad \text{$|u_j|<\frac{1}{2}$ , ...
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40 views

Estimating $p(x)$ with a nonparametric method

"Pattern Classification" book I study that in nonparametric methods we need to estimate $p(x)$, and we don't want just the averaged version of it. They give an example of theoretical procedure to ...
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2answers
956 views

Finding the mean of a uniform distribution?

I have a random set $\{a,b,c\}$ and a second set $\{e,d\}$ I draw one number first number and one from the second Letting $X_1$ denote the first number and $X_2$ the second number find, $E(X_1)$ and ...
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214 views

Poisson Process with two Lambda

I am quite stuck here, but well I am trying my best. Question: The Web page of a teacher receives hits from his students according to a poisson process with rate λ=10 visits/day . Also, his ...
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181 views

Maximum Likelihood Estimation for $\theta$ in R

I need to find estimate for the unknown $\theta$ with MLE using the R language if i'm given the following sample: $2, 2, 2, 2, 2, 8, 8, 8$. The sample is obtained from a discrete distribution with the ...
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2k views

What is an approximation for Poisson binomial distribution?

I am looking for an approximation for Poisson binomial distribution: The Poisson binomial distribution is the discrete probability distribution of a sum of $n$ independent Bernoulli trials. you can ...
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1k views

Prove there exists no uniform distribution on a countable and infinite set.

Can anyone help me with this problem, I can't figure out how to solve it... Let $X$ be a random variable which can take an infinite and countable set of values. Prove that $X$ cannot be ...
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1answer
158 views

Confidence intervals problem - nonnormal population?

this is the problem: we wish to estimate the mean execution time of a program. the program was run 35 times on randomly selected inputs, and the sample mean and the sample standard deviation ...
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1answer
804 views

What is a good distance-metric to compare 2 histograms of similarity?

in a project I want to compare 2 Histograms of similarity. My problem is to choose the right metric/ distance function. Can someone tell me, what's the difference between the Bhattacharyya distance ...
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187 views

Probability of a matrix having determinant zero

What would be the probability of matrix having determinant zero out of all matrices with all entries being positive? How does one calculate such? Edit: Restriction to natural numbers and size of $n ...
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279 views

Ratio of standard deviation to mean expressing degree of randomness

I have run many trials with random data (to compare with a real set) and found that about 90% of items occur within one-third of the mean which suggests that where the size of SD approaches the mean ...
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193 views

What is WHSSETIT?

I am working through a 7 step statistical procedure for hypothesis testing. Step 7 has a field marked WHSSETIT?. What does this acronym stand for and what is the ...
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1answer
178 views

Covariance Matrix

I'm studying Pattern Classification and there are some terms that I'm not familiar with. first one is Covariance Matrix. Say I have database of K samples, each one is a length d feature vector. ...
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1answer
347 views

Why does maximum likelihood estimation work the way that it does?

I'm wrapping my head around MLE right now and there's something about it that bothers me, irrationally I'm sure. I believe I understand the procedure: essentially we hold our observations fixed and ...
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52 views

Calculating the variance of the a regular expression assuming position dependent

Hello I am having trouble with a slightly biological problem I am given a regular expression $- [RK]-[LV]-[DE]-x(2)-Y$ this expression means that there is a string with the first position being an ...
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1answer
40 views

Distributions of sampling statistics problem (not a hw problem)

for the following question: the lifetime of a particular brand of cutting blade has a mean of 40 hours and sd of 8 hrs. when one blade fails, it is immediately replaced by a new identical blade. ...
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33 views

How to re-parametrize posterior function?

How can I write the unnormalized posterior $ f(p_1, p_2 | Y) = (z_1-1)*log(p_1) + (n_1-z_1-1)*log(1-p_1) + (z_2-1)*log(p_2) + (n_2-z_2-1)*log(1-p_2) $ in terms of the log odds-ratio $\alpha$ and the ...
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161 views

Finding maximum likelihood to $\mu$

I'd really love your help with this one. In a survey the probability that a person wouldn't like to answer on it is $p$. The survey company calls people until it meets a person who doesn't like to ...
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1answer
139 views

How is Levy distribution created?

I want to know why and how Levy distribution equation was obtained. What characteristics does it have that makes it necessary or useful?
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414 views

Name of “De Moivre's equation”

This article refers to something it calls "De Moivre's equation": $$\sigma_{x} = \sigma / \sqrt{n}$$ Basically, it relates the observed variance in a sample size to the actual variance of the ...
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49 views

Help with Statistical Analysis

I have been asked to conduct a small survey and then analyse the results of the said survey for a class project. I decided to conduct an anonymous survey of voters for the 2012 presidential elections. ...
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82 views

Determining Sample Size - Design Problem

I have a hypothetical problem were say I have a sample size of about 5000 people. I then will collect data on them say, how man calories of food they eat per day for a month. Now I can't obviously run ...
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5k views

Statistics, calculating mean and standard deviation

IF the mean of a set of scores is 80 and standard deviation is 10, explain the effect on mean and standard deviation in case following changes are made to each of the scores 1) adding 5 2) ...
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52 views

Practice question…

Harvey the clumsy waiter is in trouble at work because he has been breaking too many dishes. Suppose that the number of dishes he breaks follows a Poisson distribution with a rate of 0.11 per hour. ...
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66 views

Convergence of a running average

I'm trying to understand convergence in probability and have a specific problem; how do you show that the sample mean of n random variables $(X_k)$, each of which is the mean of $Y_{k-1}$ through ...
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2answers
305 views

Calculate odds of getting all hight point tiles in single game of scrabble

I am wondering how to best calculate the probability of getting the all of the following "high point" tiles in a single game of scrabble: Z, Q, J, X, and K In a game of scrabble, there are 100 tiles ...
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437 views

Bus stop probability question

People arrive at random times and independently at a bus stop and wait for the bus to arrive. The bus arrives at this stop once every hour. Thus, the waiting times of the people follow a uniform ...
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1answer
166 views

Notation random variable

I sometimes read the notation $$P[X \in d\omega].$$ I don't know what it means, how to pick the elements, the random variable is mapped to, to a "set" (which is none), or more precisely a variable ...
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68 views

Does second moment ergodicity imply ergodicity in the mean?

I think so, ie if the sample covariance converges to the real covariance, then the sample mean also converges to the expected value. But I couldn't get a nice proof. Also, I'm not sure if it holds in ...
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128 views
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203 views

Finding a distribution

What I want to know is, how does one go about finding the distribution that fits a particular set of data? I searched, but found nothing really satisfactory. Do we have any historical anecdotes which ...
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1answer
913 views

Likelihood function and Posterior Probability

Cited from wikipedia The likelihood function $L(θ|x)=f(x|θ)$ is not the same as the probability that those parameters are the right ones, given the observed sample. Attempting to interpret the ...
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545 views

Empirical distribution vs. the true one: How fast $KL( \hat{P}_n || Q)$ converges to $KL( P || Q)$?

Let $X_1,X_2,\dots$ be i.i.d. samples drawn from a discrete space $\mathcal{X}$ according to probability distribution $P$, and denote the resulting empirical distribution based on n samples by ...
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2answers
300 views

Relationship between variances in perfect correlation

I have two random variables $X$ and $Y$ with mean and standard deviation $(\mu_1,\sigma_1)$ and $(\mu_2,\sigma_2)$ respectively. I know that for perfect correlation the relationship is given by a ...
3
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179 views

Expected number of cumulative distinct values when sampling with replacement from a changing population over time

I'm trying to estimate the cumulative number of distinct values when sampling with replacement from a changing population of integers over time. Concretely (and forgive my awful notation here), I'm ...
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1answer
35 views

Confusion on simple probability concept

If: \begin{align} Y&=h(z,\theta)+\epsilon\\ \theta&\sim \text{Unknown Distribution}\\ \epsilon &\sim N(0,\sigma^2) \end{align} My book states: $$ ...
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1answer
398 views

Probabilities for unknown finite population from sample?

If i have a known population ($N$ marbles of which $M$ are black) and draw $n$ samples without replacement the probability to draw $x$ black marbles is given by the hypergeometric distribution. Is ...
3
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1answer
88 views

Estimate population size based on first repeat

http://mathoverflow.net/questions/14964/estimate-population-size-based-on-repeated-observation asks the following question. I take the bus to work every day. Every bus has a serial number, but ...
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192 views

What is the reason behind conditioning on an ancillary statistics?

What is the reason behind conditioning on an ancillary statistic ? Why? Why do we want to condition on an ancillary statistic ? Is it because by conditioning on an ancillary statistic, we can ...
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255 views

Is percentile affected by extreme values?

A coworker asked me how to calculate a 25-percentile and I gave him an answer but then I got unsure if I figured correctly. The problem is that our sample size will tend to be quite small so ...
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28 views

Can you determine the impact a sub-set of data has on the overall nth percentile?

Can you determine the impact a sub-set of data has on the overall nth percentile? For example, imagine I am studying the time it takes to deliver a baby. I have data from hospitals spanning the month ...
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1answer
615 views

A question on the mean and expected value

$M$ is the possible mean (average) score of a class of $n$ students and $E$ is the expected score of a student. Can someone please tell me the relationship between these two?
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1answer
284 views

Find correlation of x and y, given E(Y|X) and E(X|Y)

Suppose that X and Y are random variables such that E(Y | X) = 7 - (1/4)x and E(X | Y) = 10 - Y . Determine the correlation of X and Y . Edit: So far I've got E(x)=4 E(y)=6 Now I'm trying to ...
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78 views

Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
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1answer
26 views

Normal distribution, propability of cost.

Problem: b) http://cl.ly/image/3a3l3142432M Attempt: I find the cost to be 458 000 $. See calculations in the link and/or my thoughts here: Let cost be $c$. $F_Z(\frac{c-500000}{50000})=0.2$ and ...
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2answers
104 views

Question Regarding Poisson and probability.

i found this interesting question on the web but i am not quite sure if my solution is accurate. Honestly i would appreciate few opinions. Given Question: At a subway station, eastbound trains ...
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1answer
379 views

Iteratively Updating a Normal Distribution

Is there a way to update a normal distribution when given new data points without knowing the original data points? What is the minimum information that would need to be known? For example, if I know ...
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24 views

Finding odds of wire ratings with standard distribution?

A supplier for a factory produce wires with actual population rating of 0.6. Given the wire rating has standard normal distribution, what are the odds for a sample of 52 having an average rating of ...