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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elementary t-test question

Q: A tire manufacturer wishes to compare the tread wear of tires made of a new material with that of conventional material. 10 Cars are driven 40,000 miles as the sample set. the following data is ...
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158 views

NFC SuperBowl coin toss hot streak --> hypothesis testing and power calculation

There are many Q&A's on SE related to coin tossing - the simplest stochastic process. My Q is about relating mathematics and statistics to what in biomedicine and healthcare is termed "evidence" ...
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183 views

Ranking probability problem

$A, B, C$ are independently sampled from an uniform distribution in $[0, 1]$. We know $P(A > B) = 0.7, P(B > C) = 0.6$, what is $P(A > C)$? Is this a well defined problem? Does it have a ...
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2answers
3k views

Probability that X<Y for X and Y with known mean and standard deviation

I have physically measured two random variables X and Y and determined their respective mean and standard deviation. Both variables have a gaussian distribution. Now I wish to calculate the ...
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513 views

Scale invariance of power law distribution.

On wikipedia i have find this statement: ...it is scale invariant, and the only continuous distribution that fits this (scale invariance) is one whose logarithm is uniformly distributed. how can ...
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589 views

Distribution of a squared norm of related multivariate normal distribution.

For $i=1,2,\cdots,2^m$, let $v_i$ be dependent random variables. Suppose for $n$ large, the vector $\mathbf{Z}_n=\left(Z_1^{(n)},\cdots,Z_{2^m}^{(n)}\right)$ with ...
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1answer
139 views

Dealing with probability density functions

Let $X$, $Y$ be independent random variables with the common pdf \begin{eqnarray*} f(u) &=& \left\{\begin{array}{ll} u\over2 & \mbox{for } 0 < u < 2\\ 0 &\mbox{elsewhere} ...
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2answers
965 views

Find all possible numbers with specific mean, median and range

Given that there are $20$ positive numbers and that the median is $42$, mean is $46$ and the range is $35$. An important condition, I missed: A number can appear at the most two times. This was ...
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141 views

Variance of function-valued random variables

I am struggling with abstract definitions of basic statistical concepts. For a random variable $X$ which we assume to live in a real Hilbert (or even Banach) space $\mathcal{H}$, its expectation is ...
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117 views

Calculating minumum representative sample

Let's say I played 500 hands in poker and earned 345 units ( dollars not, important for calculation). So on long term I'm earning 345/500 = 0.69 * 100 = 69 every 100 hand session. But 500 hands is ...
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2answers
113 views

How to take derivative of this likelihood function?

I am working with the probability likelihood function $$ \log \prod\limits_{i=1}^{n} x_i^{y_i} + \log \prod\limits_{i=1}^{n}\left(1-{{x}_{i}}\right)^{n_i-y_i}. $$ I want to take the derivative with ...
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3answers
91 views

Trying to compute a balancing coefficient for evaluating load balancing in server farm

I tried to find a way to compute a coefficient varying from 0 to 1 where 0 mean perfect balance and 1 is the worst unbalance (all request go to one server). Here a practical example... imagine a ...
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1answer
129 views

Question on $p$-value for two-sided test

I'm doing some revision here and I think one of the answers in my notes is wrong. It says on my notes the answer is D). Here's the question: A researcher conducted a large sample two-sided test of ...
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0answers
87 views

Good reference on sample autocorrelation?

I'm not a statistician but I'm writing my thesis on mathematical finance and I think it would be neat to have a short section about independence of stock returns. I need to get better understanding ...
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1answer
322 views

Evaluating 'combinatorial' sum

Help me please to calculate the following sum. I have seen such kind of formulas in the papers related to combinatorics, specifically 'trees'. I am curious how to calculate or approximate this sum: ...
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343 views

Example questions for bias/consistent estimators.

Now, I know how to do this particular one, but I was wondering if anyone had a place to get questions similar to this? I've tried googling, but have not come up with anything. Consider the regression ...
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1answer
75 views

Mean of a practical distribution

I have a graph with an asymmetrical distribution (spectral response for some sensor). The graph is plotted as efficiency values versus vavelength. I must determine the median wavelength. Help please, ...
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1answer
150 views

Calculate $\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n\choose i}$

It is known that $\sum_{i=1}^n {n \choose i}=2^n$. I am wondering what would be the sum if we change the upper limit to $\sqrt n/2$, i. e. How to calculate$$\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n \choose ...
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98 views

Covariance and variance

Within the context of simple linear regression, I came across this: $$\hat{\beta}=\frac{\sum y_nx_n}{\sum x_n^2}=\frac{cov(xy)}{var(x)}$$ where I assume $cov(x,y)$ means the covariance between $x$ ...
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1answer
217 views

Application for the random variable

I am wondering if anybody knows some application for the random variable $x$ satisfying the following condition: for $t\ge 1, a>0, c>0$ $$ P(|x|\ge t)\geq \frac{c}{t^a} $$ It looks like thhis ...
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1answer
2k views

Logistic Sigmoid Function with a vector input

For a statistical learning problem (classification), I have the data set $\{ (x_i,y_i) \}_{i=1}^n$ with $x_i \in \mathbb{R}^2$ being the input data and $y_i \in \{0,1\}$ the possible classes. The ...
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Finding $E\left[\frac{\sum_{i=1}^n X_i^2}{(\sum_{i=1}^n X_i)^2}\right]$ of a sample of gamma random variables

Suppose $X_1,\ldots,X_n$ is a random sample from the $\Gamma(k,\lambda)$ distribution where $\lambda$ is unknown and $k$ is a positive integer and known. How can I find $$E\left[\frac{\sum_{i=1}^n ...
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2answers
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Expected value of $\ln X$ if $X$ is $\Gamma(a,b)$ distributed.

I'm new here and hope you can help. It's really late here in South Africa, maybe my mind just doesn't want to function now! But I need to figure out how to get a closed form expression hopefully for ...
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2answers
184 views

How to prove a random variable taking values in $[0,1]$ range has variance no larger than $\frac{1}{4}$?

How can I prove that a random variable taking values in $[0,1]$ has variance no larger than $\frac{1}{4}$? If it matters, discrete and continuous proofs are both welcome.
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2answers
148 views

How to find the number of unique sets of 7 letters?

The order of the letters does not matter, so: ABALNKM is the same as ALMKNBA bonus points How would I determine the number of sets where any letter can only be repeated a maximum of 4 ...
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1answer
110 views

Why does this probability equivalence of events hold?

$P(X_0 = j, X_m \ne j, 1 \le m \le n-1) = P(X_m \ne j, 1 \le m \le n-1) - P(X_m \ne j, 0 \le m \le n-1) $ Where $\{X_n\}$ is an irreducible Markov Chain with a finite state space.
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1answer
328 views

Calculating kWh from multiple watt readings

I have a file which contains watt readings for various times throughout the day. A reading is stored roughly every 6 seconds. ...
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2answers
297 views

finding MLE of parameter $\theta$

suppose $X_1,X_2\ldots,X_n$ be a random sample of distribution with probability density function $$f(\theta, x) = \begin{cases} \theta &\text{if } x=-1 \\ (1-\theta)^2 \theta^x & \text{if } ...
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1answer
108 views

$\chi^2$ critical value ranges

For reference, here is the $\chi^2$ distribution table for degrees of freedom = 7: p 1% 5% 95% 99% v=7 1.239 2.167 14.07 18.48 ...
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138 views

Information and MLE

Is the asymptotic distribution of the MLE of an unbiased estimator $N(\theta, \frac{1}{nI(\theta)})$? So if you know the Cramer-Rao lower bound, you know the asymptotic distribution of the MLE of an ...
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1answer
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Statistics : Where did this degree of freedom formula for the T distribution come from?

I am on the hypothesis testing for two populations unit. I need some intuitive explanation as to why this formula is used. My statistics professor put this up on the board but he didn't explain why ...
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1answer
223 views

expected value question

Can we say that $$E \left(\frac{n}{n-1}X \right) = E(X)$$ because $n/(n-1)$ is basically equal to 1. Or can we not say this? I am just factoring out the n/(n-1). $n$ is the sample size.
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1answer
267 views

Normal distribution problem

Hello Mathematics dep! Variate $X$, which follows the normal distribution, has median $\mu = 14$ and variance $\sigma ^2 = 9$. What are the odds that $X > 12$. Attempt at a solution: $P(X>12) ...
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1answer
750 views

Describing about the probability distribution of a scenario

The relative frequency distribution of the weights of all passengers using an elevator is mound-shaped, with mean $150$pounds and standard deviation $35$pounds. Suppose there are n passengers in the ...
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1answer
177 views

Question concerning a possible constant sum of squares in a circle

Assume a circle, of some radius $r$ say $r=1$. Consider a fixed $n$, say $n=30$, of points on the circumference of the circle. That $n$ points $(x_k,y_k)_{k=1,\ldots, n}$ define a mean in the interior ...
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3answers
183 views

An equation that occurs in ecology

In the field of ecology, a well known relation between the number of species and the size of an island can be approximated by a power function of the form: $S = c A^z$ $S$ = number of species $c$ = ...
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1answer
85 views

Help with Bernoulli Trials and continuum of agents / CLT

Suppose I have a continuum of agents $i \in [0,1] $ where each agent i takes the action $x_i$ where $x_i = 1$ if $ \epsilon_i >-a$ and 0 otherwise Assume that $ \epsilon_i $ has a standard normal ...
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0answers
253 views

normal random variable distribution

i have such problem in the book of Applied statistic and probability for Enigneering and need some help to solve it.problem is following: Let random variable X denote a measurement from a ...
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1answer
366 views

Gauss-Markov estimator properties

Consider a linear model $$ y = Ab+n, $$ where $b \in \mathbb{R^m}$ is a parameter to be estimated, $n \in \mathbb{R^{n}}$ is a noise with mean $\mathbb{E}n = m_{n}$ and with covariation matrix ...
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0answers
156 views

Understanding Linear Regressions with Least Squares

I am currently trying to understand the linear regression fit by least squares for my machine learning homework, where I implement it and have to plot the result: I have given two data sets, ...
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1answer
251 views

Probabilities and Sum of Normal Distributions

Suppose I have: $ V_1 = X_1+A $ where $A>0$ is some constant $ V_2 = X_2+B $ where $B>0$ is some constant Furthermore assume that $X_1$ and $X_2$ are independent and distributed with a ...
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1answer
86 views

Cluster Analysis Terminology question

Let's say that I have a set of datapoints. Let's say I also have a set of pairs of these points, where a single point can be in multiple pairs. Let's say I also have a set of triples, etc with the ...
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1answer
204 views

Poisson Distribution

I've a question here that I may possibly help with: "Suppose that the number X of errors in an assignment submitted by each student in a certain group is a random variable that has a Poisson ...
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1answer
174 views

Finding two populations in a set of data

I have a set of data (metallicity of globular clusters) and wish to determine the presence of two sub-populations. I know there are two populations but I am unsure on how to split the two. I have ...
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1answer
8k views

In what situations should I use and not use a pooled estimator for $\hat{p}$

In a question, it says that a true-false exam is used to discriminate between well-prepared students and poorly prepared students. There are $\frac{205}{250}$ well-prepared students and ...
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1answer
287 views

Degrees of freedom in a Multivariate t Distribution?

Can I have different degrees of freedom for each dimension in a Multivariate t Distribution? The functions that I use in Matlab (mvtpdf) and Mathematica (MultivariateTDistribution) accept only one ...
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464 views

fit a skewed data set with skew normal dist using excel

I want to fit a skew normal curve to a data set in excel. My goal is to model incomes in geographic areas in order to get the # of people < $23,000 income, using several median values and their ...
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2answers
3k views

Statistics: Could someone show why this exponential pdf integrates into this particular cdf

I have the following exponential distribution: $$f(\lambda, x) = \begin{cases} \lambda e^{-\lambda x} &\text{if } x \geq 0 \\ 0 & \text{if } x<0. \end{cases}$$ I need to show that this ...
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1answer
124 views

Calculating the regression equations

I have four data points $(1,2), (2,4), (3,5), (5,7)$ and Im looking for the least squares regression line that best fits them. I use the normal equation $A^tAx=A^tb$ in this form - ...
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1answer
70 views

limit distribution $Y_n=\frac{1}{n}\sum\limits_{i=1}^n (X_i-10)^2$

Suppose $X_1,X_2,\ldots,X_{n}$ be a random sample of $B(20,.5)$. How can find limit distribution $$Y_n=\displaystyle\frac{1}{n}\sum_{i=1}^n (X_i-10)^2$$