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 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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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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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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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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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
313 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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439 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
167 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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69 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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209 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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929 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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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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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 ...
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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
403 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 ...
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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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1answer
197 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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262 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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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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621 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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287 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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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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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
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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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391 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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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 ...
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191 views

Integral of a Gaussian distribution over the unit ball

Given a multivariate normal $X \sim N(\vec{0},\Sigma)$, I would like to calculate the pdf when sampling from the unit ball $(||X||_2=1)$. Specifically what is the value of the normalizing factor $Z$ ...
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1answer
144 views

Bounding the standard deviation of a bounded random variable

Suppose $X$ is a random variable (discrete or continuous) whose values lie in the segment $[0,1]$. Is it safe to say that the standard deviation of $X$ is between $0$ and $\frac{1}{2}$?
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Is there a notion of reliability for percentiles?

Say I run an experiment measuring some parameter A on a sample group of a population where A is a non-negative real number. Is there some way of knowing if my percentile results are reasonable? ...
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617 views

Jacobian Transformation for distributions

Consider random variable $X$ and its distribution $f = \dfrac{3x^2}{2}$ where $-1 < x <1$ What is the distribution of $Y= |X|$? What about $V=X^2$? And $W=X^3$? I basically did the ...
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1answer
127 views

Discrete Random Variable of a die

A icosahedral die has each face marked with different numbers from 1 to 20. Assume that each face is equally likely to occur on a single roll, the outcome is a random variable $X \sim DU(20)$ 1) If a ...
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calculating correlation

If one fair six-sided die is rolled, suppose that $X$ is the total number of even numbers shown and $Y$ is the total number of fives shown. How can I go about calculating the correlation exactly in ...
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How to generate a continuous random variable with the density $f(x) = 1.5x^2, -1 < x < 1$?

Let $U$ be a Standard Uniform random variable. Show all the steps required to generate a continuous random variable with the density $f(x) = 1.5x^2$, $-1 < x < 1$. I'm not looking for exact ...
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How to assure quality control to a 4% level of significance based on a 200 item sample with 24 defects?

For quality control purposes we have to accept or reject a large shipment of items. In order to do so we collect a sample of 200 items, 24 of which are defective. The manufacturer claims that at most ...
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82 views

Definition of exponential distribution and its relation to Poisson

I want to understand the Poisson and exponential distributions correctly. Would this be correct "If $X$ follows a Poisson distribution, then $T$ measures the probability that you have to wait $t_a$ ...
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Difference between Chi-Square Test (goodness of fit) and binomial?

This is probably an unnecessary question, but how would I know given a problem when it's appropriate to use a chi-square test vs. binomial or normal distribution in a problem? Let's say that the ...
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What does it mean to say a random variable is non-negative?

How would you define a random variable to be non-negative ??? What are some examples of a Negative random variable ???
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Linear relationship of a company's profit

Assume a linear relationship for a company that has several shops is not known. Let $Y_i$ be the profit the shop number $i$ makes in the coming year. Let $x_i$ be the size of the shop number ...
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272 views

Hypothesis testing for Variance - choosing a null hypothesis.

A gun-like apparatus has recently been designed to replace needles in administering vaccines. The apparatus can be set to inject different amounts of the serum, but because of random ...
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What is the distribution of the sum of absolute differences?

Suppose there is a $2\times2$ matrix $O$ of observed values: $$O = \begin{bmatrix}o_{11}&o_{12}\\o_{21}&o_{22}\end{bmatrix}$$ and two matrices $E_1$ and $E_2$ of expected values: $$E_1 = ...
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Why can we interchange summation sign and Variance?

Why can we interchange the summation sign and variance sign? $$\mathrm{Var}\left(\sum^n_{i=1} Y_i\right) = \sum^n_{i=1} \mathrm{Var}(Y_i) $$ Is there a proof for this?
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Confidence interval for a normal mean when the variance is unknown

For this problem , how to calculate when $t.025,8 = 2.306$?
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Statistics: Combination problem

Good day everyone. I'm doing a combination problem and its solution keeps eluding me. Problem: Of a hand of 13 in a deck of 52 how many combinations are there of none of the cards being greater ...
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Is it possible to calculate the mean and standard deviation from a median and quartiles?

Any advice would helpful. I understand that the reporting of median and quartiles for small samples is an indication of skewed data. If such is correct, then is it useless to try to work out the mean ...
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bernoulli trial

I have to find a probability that 2 in 30 electrical component coming off a production line is defective. The probability to get a defective one is 0.04. Assume that whether or not a component is ...
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Expected value for $f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$

$$f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$$ where $0<x<\beta$. So these are three terms all multiplied ...
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How to find $E(X|S)$ where $S=X+Y$ and $X$ and $Y$ are independent random variables with some density function $f$

So far I have this, but I am not sure if you are allowed to do this or if it is correct: I said since $X$ and $Y$ are independent and follow the same distribution, they must have the same ...
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Generating statistics from data samples, that are of snapshots.

My question is part math part computers but I need more of the math answer then a computer answer. I am going to be grabing snapshots of a moment of time on how a computer is performing. Because of ...