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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50 views

How I figure what (theoretical) dice are needed to achieve a certain curve?

I am posting here by suggestion of RPG.se I want to make character stats that fit within certain bell curves depending on choices during character creation (for example race, gender, class, sprokets ...
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4answers
92 views

Displacement law: $ \sum_{i=1}^{n} (x_i - \bar{x})^2 $ why is it squared?

$ \sum_{i=1}^{n} (x_i - \bar{x})^2 $ where $ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $ (mean value) I would like to understand why the term inside the sum gets squared. Why isn't it enough to e.g. ...
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1answer
1k views

For the binomial distribution, why does no unbiased estimator exist for $1/p$?

Suppose that $X \sim \mathrm{Binomial}(n,p)$ for $0 < p < 1$ Why does no unbiased estimator exist for $1/p$? My approach: We try to find the structure of $E_p(U(x))$, where $U(x)$ is any ...
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2answers
163 views

Statistics-Math-Probability

An urn is filled with 10 green balls, 4 red balls, and 2 orange balls. Three balls are selected without replacement. Calculate the following probabilities: (a) P(at least one ball is red) I think ...
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154 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
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1answer
47 views

Regression towards the mean?

I am upset with what examples testing "regression to the mean" seem to allude to: People claiming to have "ESP" take a test, and A's score was 2 standard deviations above the mean, whereas B's score ...
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1answer
39 views

Statistics binomials

A recent medical survey reported that 45% of the respondents to a poll on patient care felt that doctors usually explain things well to their patients. Assuming that the poll reflects the feelings ...
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1answer
80 views

Meaning of off-diagonal multivariate covariance matrices

My terminology might be a bit sloppy. I apologize in advance. I'm reading on multivariate probabilistic distributions, particularly on Gaussian normal distribution (in the context of probabilistic ...
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1answer
129 views

Robust Standard Errors

For OLS, my professor said that you should always test for heteroscedasticity first, rather than going straight to the adoption of robust standard errors. I didn't quite follow this and no ...
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1answer
333 views

Correlation Coefficient Distribution Function: An Apparent Discrepancy?

I'd like to explain an apparent discrepancy between: (1) The sample correlation distribution function between sample vectors for a bivariate, correlated random variable (correlation coefficient = ...
2
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0answers
83 views

How many different ways can 10 octupuses touch legs?

There are 10 octopuses (octopi?). Each octopus has 8 legs. Legs on an octopus can only touch touch legs on other octupuses. Assuming each leg touches exactly 1 other leg, how many different ...
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0answers
102 views

Expected minimum of a finite random walk.

So I couldn't find any resource for how to calculate the expected minimum of a random walk. Since it is such the minimum of the random variables are actually not independent as they are cumulative ...
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1answer
45 views

Variance with correlated variables

A simple question that I don't manage to solve: I can use different methods to measure a magnitude $x$. The results of these methods are correlated and have some uncertainties. Combining the results ...
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0answers
45 views

What is Lagrange's Identity in terms of Covariance and Variance?

This is Lagrange's Identity What I've to put in the middle of this relation instead $\color{red}{?}$, which led to find Lagrange's Identity in terms of variance and covariance? $$\lvert ...
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1answer
329 views

Why isn't a t test used when comparing two proportions?

All the examples I've seen say to use a z test to compare two proportions. For example, n=13, x=0.22 versus n=10, x=0.44. Then all the examples warn that the z test doesn't work with low sample sizes. ...
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1answer
55 views

Differentiability of linear least squares

Show that least-squares $\|y-X\beta\|^2$ is twice differentiable and has minimizer. I understand that the second derivative is $X'X$. Also it is a composition of linear function which is ...
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1answer
170 views

Stat question about Markov Chain

An urn contains five red and three green balls. The balls are chosen at random, one by one, from the urn. If a red ball is chosen, it is removed. Any green ball that is chosen is returned to the urn. ...
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1answer
913 views

Standard error of Method of Moment estimator

Suppose that $X$ is a discrete random variable with $P(X=1)=θ$ and $P(X=2)=1−θ$. Three independent observations of $X$ are made: $x_1=1,x_2=2,x_3=2$. Find the MOM estimator of $θ$. What is ...
0
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1answer
50 views

Guessing the sex of my app user based on the overall app downloads statistics

I have got an app that has 72% male, 22% female, and 7% unknown users. In order to make more money from displaying ads, I want to pass the sex of the user to the ad server. How close would I be if I ...
0
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1answer
60 views

Question of Normal Distribution ??

I do some exercise, and there is an interesting question, can anyone can help me solve it. suppose that the heights in inches of the women in a certain population for a normal distribution with mean ...
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2answers
136 views

maximum likelihood estimation problem tutorial

The question is find the δ by The maximum likelihood estimation? My answer is δ=0 but I am not sure whether it is correct and how tho show its biasness?
2
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1answer
71 views

sampling distribution with no continuity correction

Assume that the distribution of the height of a plant is normally distributed with a mean height of $\mu = 83.4$ inches and standard deviation of $\sigma = 9.1$ inches. Find: if three such plants ...
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0answers
85 views

Alternate form of Cramer-von Mises type statistic

When I read a paper from Rothman and Woodroofe (1972) for testing symmetry, they used the following form: $$ R_n = \int_0^1 n \left[\hat{F}_n (x) + \hat{F}_n (-x) - 1 \right]^2 dF_n$$ where $ ...
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1answer
73 views

How to calculate two asset portfolio (problem finding the Population correlation coefficient)

I have two assets: A has an expected value of $12$ % and a standard deviation of $8$%. B has an expected value of $15$ % and a standard deviation of $12$ %. Suppose that we invest $75$ % in A and ...
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1answer
543 views

probability continuity correction

consider taking a random sample size of 25 from a population in which 42% of the people have type A blood. what is the probability that the sample proportion with type A blood will be greater than ...
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2answers
317 views

How leafy are my plants?

Suppose a botanist grows many individually potted eggplants, all treated identically and arranged in groups of four pots on the greenhouse bench. After 30 days of growth, she measures the total leaf ...
2
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1answer
47 views

Disagreement about rejection region for upper-tailed hypothesis test

If we look at the solutions for the first problem here and the first problem here, we see both problems are one-tailed tests for the upper-tail. However, in the first paper the rejection region is Z > ...
2
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1answer
60 views

How to calculate an expected value from a cluster of results

I have a vector of points r where each element has an x and y value. For every one of those points there is a z value, which corresponds to a measurement taken at position r. If I have say n ...
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2answers
2k views

X follows an exponential distribution, calculate Expected value of sqrt(X).

Problem: Let X follow an exponential distribution with expected value of 1. Define Y=sqrt(X). Calculate E(Y). This is my first course in probability theory (5 weeks ≈ about 5*40 hours of workload) so ...
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2answers
87 views

Variance of difference of two sample means from the same population

We take two samples from $Z$ of size $n_1$ and $n_2$ and take the difference of the mean of these samples. Both should have the same expected value, so the mean is zero. But what is the variance of ...
2
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1answer
439 views

Differentiation for least squares method?

Is there any reason that we use mathematical differentiation of least squares method for regression analysis? The theory say we use differentiation supposing the sum of errors is 0. I I don't really ...
0
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1answer
39 views

In Bayesian approaches, comparing the approximated distribution and the true distribution

I'm asked to compare the distance of the approximated distribution and the true distribution in a Bayesian approach. While we used Laplace approximation to find the MAP of the target posterior ...
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7k views

How to calculate relative error when true value is zero?

How do I calculate relative error when the true value is zero? Say I have $x_{true} = 0$ and $x_{test}$. If I define relative error as: $\text{relative error} = \frac{x_{true}-x_{test}}{x_{true}}$ ...
2
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2answers
124 views

Question about the logic behind hypothesis testing

Let us say that we have this following problem: "A government agency claims that more than 50% of US tax returns were filed electronically last year. A random sample of 150 tax returns for last year ...
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1answer
72 views

Buying the best hard drive, or: Confidence interval for difference between means, variance of ordinal data, etc.

I think my hard drive is on the fritz, and so I've been sorting through my options on Amazon. However, the ratings on Amazon don't really make distinctions as nicely as I'd like; the only readily ...
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2answers
626 views

Covariance of minimum and maximum of uniformly distributed random variables

Let $(X_1,X_2,\ldots,X_n)$ be iid, such that each $X_i$ has the uniform distribution on the interval $(a,b)$. Calculate $Cov(\min(X_1,\ldots,X_n),\max(X_1,\ldots,X_n))$. The task seems very hard to ...
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1answer
45 views

Calculate conditional expectation

If $N\geq2$ and consider a population consisting of the first N positive entire numbers: {1,2,...,N}. Let $$\mu=\frac{1}{N}\sum_{i=1}^{N}i = \frac{N+1}{2}$$ be the population average and ...
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1answer
466 views

Why is the expected value of the squared value equal to the sum of the standard deviation and the mean?

I am currently reading the proof here that one has to divide by $n-1$ in order to get an unbiased estimator of the population variance. What I do not get is this: \begin{equation}E[y_i^2] = \sigma^2 ...
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2answers
46 views

Proof using binomial coeff

I don't understand the step between left side and right side of my ? I
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1answer
251 views

Price of a n toss coin game

I was thinking of a game where a player pays one dollar to toss a coin. If the result is a head they get their 1 dollar bet plus an additional 1 dollar. If the outcome is a tails they lose their 1 ...
0
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2answers
38 views

Statistics: If $X_1$ and $X_2$ are both normally distributed then explain why $X_1 - X_2$ can be standardized with mean 0 and standard deviation of 1

I am currently studying hypothesis testing for two populations and I would like a math major or someone experienced to explain to me why this particular statistic has a mean of 0 and a standard ...
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0answers
92 views

standard deviation of a rate of phone calls

I am trying to implement some real time monitoring. I have a real time data feed from which I can calculate the number of phone calls received in the last hour (now-60 minutes). I want to alert when ...
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1answer
34 views

Events occuring

Let $A_1,....A_{20}$ be mutually independent events and let $p_i$ be there probabilities $(p_i = P(A_i))$ , $i = 1,2,...,20$. Express $Pr$(no events occuring) in terms of $p_1...p_{20}$ Express ...
2
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3answers
991 views

Expectation of minimum of normally distributed random variables

Let $(X,Y)$ be normally distributed and such that $\;\;\;\;\mathrm{Cov}(X,Y)=\varrho$, and $\mathrm{Var}(X)=\mathrm{Var}(Y)=1$. For which $\varrho$ does the following equality hold? $\;\;\;\;\min ...
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2answers
2k views

Probability of exactly one defective item in a sample of three…

In order to determine the quality of a shipment of 20 parts, a sample of 3 items is randomly selected without replacement from the shipment. Four of the 20 items in the shipment are actually ...
3
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1answer
500 views

What is the expected number of k-length streaks in n rolls?

Given $n$ flips of a coin with success probability $p$, what is the expected number of $k$-length win streaks in $n$? (I've looked for this question online, but the answers always restrict $n$ to be ...
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2answers
45 views

Proof for expected value of geometric RV without using derivates or other “fancy” methods

Is it even possible? I'm guessing it is, but I get stuck very early on: $E[X]=\sum_{k=1}^\infty kp(1-p)^{k-1}=\sum_{k=1}^\infty k(1-q)(q)^{k-1}=\sum_{k=1}^\infty k(q^{k-1}-q^k)=\sum_{k=0}^\infty ...
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0answers
200 views

Why is my sample standard deviation not an unbiased estimator?

I was goofing around with a spreadsheet and decided to show empirically that the sample standard deviation is an unbiased estimator of the true population standard deviation. Link to Excel File I ...
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1answer
80 views

Find the original pdf given conditions in order to find Maximum Likelihood Estimator

Random Sample Distribution from $X_1,X_2,...,X_n$. Given the p.d.f of $f(x;\theta)$ Find the Maximum Likelihood Estimator $\theta$; When $\theta = 0$, $f(x;\theta) = 1$ where $0<x<1$. When ...
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0answers
31 views

We said the data is normally distributed, based on the raw data or residual?

I have a confusing regarding the assumption test for the data, in some theory were said that there are three assumption of data as we called as "good" data: Independent Normally distributed ...