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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Use a Trendline Formula to Get Values for Any Given X with Excel

First off, is this question appropriate for this Stack Exchange? If not, I'll remove it. Now to my question: Is there an easy way to apply the trend line formula from a chart to any given X value in ...
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148 views

Understanding some basics

Let $x_1,\ldots,x_n$ be any numbers and $\bar{x} = (x_1+ \ldots + x_n)/n$. Then I need to prove that $$min_a \sum_{i=1}^n (x_i - a)^2 = \sum_{i=1}^n (x_i - \bar{x})^2$$ I do not need the answer but ...
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how does expectation maximization work?

I'm reading a tutorial on expectation maximization which gives an example of a coin flipping experiment (the description is at ...
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209 views

Expectancy value for the percentage of points lying in the Convex Hull (3D)

Suppose I chose n uniformly distributed random points in a 3D cube. What is the expected value for the percentage of points lying on the convex hull as a function of n? Just as a reference, I made ...
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280 views

expected value for min estimated entropy

We have a random generator that generates independent random bits with probability $P(x=1) = P$ and $P(x=0)=1-P$. Given $N$ random independent bits, we estimate $P$ by $\hat{P} = N_1/(N_0+N_1)$. ...
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164 views

How to fit fixed data from two linear functions

I have a set of points $(x, y)$ where each one comes from either one of two linear functions: \begin{align*} y &= m_1 x + b_1\\ y &= m_2 x + b_2 \end{align*} Is there a fitting method ...
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769 views

Does exceptionalism persist as sample size gets large?

Which of the following is more surprising? In a group of 100 people, the tallest person is one inch taller than the second tallest person. In a group of one billion people, the tallest person is one ...
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712 views

Distribution of the coefficient of variation of independent normal variables?

Let $X_1,\dots,X_n$ be Gaussian random variables with mean $\mu$ and variance $\sigma^2$. What is known about the distribution of $$\frac{\sqrt{\frac1{n-1}\sum_{i=1}^n(X_i-\bar X)^2}}{\bar X} $$ ...
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733 views

Odds of winning a contest

In a random drawing with 50 entries are the odds of winning better if I: A. Enter 1 contest 3 times or B. Enter 3 contests 1 time
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739 views

Understanding the Pareto distribution as applied to wealth

The Pareto distribution is used to say, given a particular person X, what is the pdf of his wealth. I would like to explore the reciprocal question: Given the total amount of wealth in a population, ...
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298 views

Stochastic Urn Process using a Pareto distribution

N urns are assigned m balls in a stochastic process based on a Pareto distribution. The process is as follows: X is a Pareto random variable (xminimum = 1, alpha is a parameter) if X > N, throw the ...
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1k views

Finding joint sufficient statistics

I'm trying to find two statistics T1, T2 such that (T1, T2) is jointly sufficient for (λ, θ) for a random sample $X_1,\ldots,X_n$ from a two parameter exponential distribution. $f(x) = \begin{cases} ...
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80 views

Equation for Ranking Top Posts

I am trying to write an equation to solve my problem of figuring out which post is most popular. While one would think the one with the highest stars would be, if the stars are equal then the one with ...
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580 views

Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
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83 views

How to estimate $SE$ from $95\%$CI, $\alpha $ and $n$?

Background I am conducting a quantitative synthesis (meta-analysis) that includes previously published data. For the model that I am using, an exact or inflated estimate of SE is acceptable. ...
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162 views

Real valued sufficient statistic

How to find the real valued sufficient statistic of $\theta$ for a random sample from a distribution with the probability density function written below? $$f(x;\theta) = \theta ax^{a−1} \exp(−\theta ...
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136 views

Notation question with Dirichlet processes

This is a question on how to make sense of an equation (specifically, a description of a distribution) that, no matter how I slice it and dice it, always has some left over pieces that don't fit. The ...
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208 views

Sufficient statistics

Suppose $T_1(X)$ is a sufficient statistic for $\theta$ for a family of distributions indexed by $\theta$ and $T_2(X)$ is another statistic that induces a “finer”(what is meant by a finer ? ) ...
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What is the difference between all types of Markov Chains?

I have been looking for some good material covering Markov Chains but everything seems so difficult to me... After reading about the subject, I figured out that there is basically three kinds of ...
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4k views

Definition of mean as an integral over the CDF

I'm reading a statistics textbook which defines the mean of a random variable $X$ with CDF $F$ as a statistical function $t(\centerdot)$, where $$ t(F) = \int x \, dF(x).$$ Can someone explain this ...
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62 views

Confidence in classifying data

If I have 5 people classify an image as offensive or not, how can I calculate my confidence that the answer that 3 or more people agree on is the correct answer? I have past accuracy rates, but am ...
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93 views

Filling an array with values which sum to $1$ and look like gaussian

I need a formula which will fill an array of X slots with values (from $0$ to $1$). The sum of all array values should be $1$. The values should look like this: ...
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510 views

Determining odds in Blackjack?

How do you determine the different odds in Blackjack? For example, what would be the difference in odds from using 1, 2, or 3 decks? Also, what would be the difference in odds if you shuffle the deck ...
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How do you find the number that corresponds to Quartile 1, when given an even number of scores?

If you have an odd number of data values {39, 40, 42, 44, 47, 48, 49, 51, 53}, what is the score that value that corresponds to Quartile 1? In general, I would like to know if you include the ...
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Likelihood with gaussian noise

In the book "Markov Random Field Modeling In Image Analysis" by Stan. Z Li, 3d Edition. Page 15. An observation d of a field f (with m nodes) with i.i.d. gaussian noise: $d_i = f_i + e_i$, there ...
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250 views

Interpretation of Markov's Inequality

Markov's inequality states that Pr(X >= t*E(X)) <= 1/t. This is great for asking "What is the probability that we get more than t times our expected value". ...
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Sufficient statistics

I'm learning about sufficient statistics and I understand basic stuff that it is the minimum information we need to represent a statistic (very vague definition, I know. I'm learning). I'm trying to ...
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1k views

Statistics: Predict 90th percentile with small sample set

I have a quite small data set (on the order of 8-20) from an essentially unknown system and would like to predict a value that will be higher than the next number generated by the same system 90% of ...
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96 views

Measure change in two slopes for a set of data

I have a dataset with yearly levels of corruption in a number of countries. $\begin{array}{c|lcr} \text{year} & \text{corruption} \\ \hline 2001 & 5 \\ 2002 & 7 \\ \end{array}$ etc. I ...
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149 views

Probabilities of cards distribution

Hw do i estimate which pattern is the most likely in the distribution of the number of cards in each suit in a hand among 5-3-3-2; 4-4-3-2 and 4-3-3-3. This is a Bridge Game related problem
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403 views

Does this induce a proper probability distribution on the space of covariance matrices?

Say I define a probability distribution on $P$ dimensional symmetric matrices such that the diagonals are strictly positive but the off diagonals are unrestricted (except for the symmetry constraint). ...
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251 views

Showing consistency of an estimator

I have $\log X \sim Exp(\vartheta - 1)$ and I would like to show $$ P \Big [ \Big |\frac{1}{\frac{1}{n} \sum_{i=1}^n \log X_i} - (\vartheta - 1) \Big | > \varepsilon \Big ] \rightarrow 0 ...
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408 views

Order statistic

I've two questions (1) How do I determine the distribution of the first order and the highest order statistic for sample of random size N is taken from the continuous uniform(0, $\theta$) and ...
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234 views

What is the characteristic formula for the addition of two lognormal distributions?

If I have two lognormal processes (X and Y) with mean and volatiilty for each and also correlation between the two, what is the characteristic formula of X + Y (i.e. what is the new mean and new ...
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121 views

Calculating the “jaggedness” of a distribution

I'm sure "jaggedness" isn't the right term to use here, so please correct me. I'm trying to quantify how jagged a distribution is. For example, this is moderately jagged: distribution #1 This is ...
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203 views

dependency between regression coefficients and probability distribution

let's consider regression problem. Given set of training data $\{(x_i,y_i)\}_{i=1}^N$, $x_i \in \mathbb{R}^n$ and $y_i \in \mathbb{R}$, find prediction function $y = f(x)$, e.g. in RBF regression case ...
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492 views

Probability and Distribution

Consider a random sample from a distribution with the p.d.f. (for some $c > 0$) \begin{equation} f(x) = cx^2 (1 − x), \text{if } 0 < x < 1; \end{equation} \begin{equation} 0, \text{ ...
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2k views

Calculate variance from a stream of sample values

I'd like to calculate a standard deviation for a very large (but known) number of sample values, with the highest accuracy possible. The number of samples is larger than can be efficiently stored in ...
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File Size Stats

This is kind of a programming question, but really it's math in disguise! I'm working on a little project that will look at the size of a file and make sure it's in within an acceptable range based ...
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187 views

Hypothesis Testing - Newton or Einstein?

After taking measurements on the deflection of stars during the 1919 eclipse in Borneo... It was found that: n = 15 Mean = 1.4824 angular units (AU) SD = 1.349237 AU SE = .348371 AU A) Test the ...
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290 views

Calculating a Positive Relative error

I need to graph the error between a set of observed values and an expected values. I calculate the relative error for each point as: ...
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238 views

Lottery chances

What's statistically more likely to win you the UK lottery (6 numbers out of 1-49), To play the lottery weekly or to save the money and buy multiple tickets all in a oner? There is a discussion in ...
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4answers
123 views

How much noise will the average of N noisy signals have?

(Inspired by this question on the photography site) Say you have N copies of the same signal, each with a layer of noise on top. You average these copies together in an attempt to reduce the effect ...
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268 views

Specific statistics problem: Ranking universities

What do you think is the best way to rank universities participating in a competition if we know for each university the number of recognitions that it's students won. Each university can participate ...
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3k views

Difference between time series and stochastic process?

What is the difference between a "time series" and a (discrete-time) stochastic process?
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171 views

Inverting $(A + D_p)$ efficiently for many diagonal $D_p$ (and $A$ SPD)

At one step of an algorithm (i=1: something big) I have to draw a vector from the normal distribution $X_p^{(i)}\sim N((A^{(i)} + D^{(i)}_p)^{-1}m, (A^{(i)} + D^{(i)}_p)^{-1})$ for $p=1:P$, with a ...
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623 views

20% in one year vs 4% per year over 5 years

I am looking for information for some extended family members. A medical procedure has a 4% chance of stroke per year over 5 years. One family member has said that that is a 20% chance of stroke ...
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721 views

Maximum likelihood and sufficient statistics

$$f_T(t;B,C) = \frac{\exp(-t/C)-\exp(-t/B)}{C-B}$$ where our mean is $C+B$ and $t>0$. so far i have found my log likelihood functions and differentiated them as follows: $$dl/dB = ...
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346 views

Probability calculation, system uptime, likelehood of occurence

A little stumped! This is probably a very basic probability question, but I am lost. At work I was asked the probability of a user hitting an outage on the website. I have some following metrics. ...
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Is positive the same as non-negative?

I would assume the answer to my question is yes, but I want to make sure because my book uses both terminologies. Please also indicate where zero falls into the mix. UPDATE: Here is an excerpt from ...