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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Expectation of squared time-scaled Brownian process

According to an article I'm studying ("Time series, self-similarity and network traffic by Mark Crovella) the expectation of the square of a time-scaled Brownian motion process $E[ B(ct)^2 ]$ where ...
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Critical exponents and point-wise convergence

A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. ...
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is the following a positive definite matrix?

Let $X_1,....,X_n$ be some positive random variables. Is $A_{ij} = \frac{E[X_i*X_j]}{E[X_i]E[X_j]}$ a positive (semi) definite matrix? (note that $B_{ij} = Cov(X_i,X_j) = E[X_i*X_j] - E[X_i]E[X_j]$ ...
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360 views

License Plate Statistics

California issues license plates in numeric order (if we turn the letters into numbers). I have fun noticing the latest plate I have seen. I am interested in what you can derive from a series of ...
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405 views

can KL divergence converge to 0, but variance of log odds not converge to 0?

Let $p(x)$ be a fixed distribution. Let $A, C > 0$ be constants. Let $\epsilon > 0$. Can we find an example of a distribution $q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\epsilon}) < \epsilon$, ...
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Average amount of money received from pulling three random coins

Let's say I have a bag of coins, which contains 1 quarter, 2 dimes, 3 nickles and 4 pennies. If I were to randomly pull out 3 coins, on average, how much money would I get?
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Basic probability… probability that two users share a key out of a keypool of 20?

Let's say I am distributing a bunch of encryption keys, where if two users have the same key, they can communicate. I have 20 different keys, and I am giving each user 2. Since encryption keys don't ...
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Algorithm for approaching a position

I'm looking for an algorithm, that can process an amount of (GPS-) positions and determine, if the actual position gets closer to a way point. What is the "best" way to filter variability in the ...
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215 views

Meaningful way to reinterpret a sleep-movement-graph/set

I am building an android application that records accelerometer data during sleep, so as to analyze sleep trends and optionally wake the user near a desired time during light sleep. I have already ...
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753 views

Recommend a statistics fundamentals book

To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away ...
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The pseudoness of pseudorandom number generators

Is there a reasonable statistic test one can do to standard random number generators (say, one of those that come built in in Python libs) which shows they are not really random? (By reasonable I ...
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Simple algorithm effectiveness (Least-Squares Regression Line in Terms of Sample Variances)

I hope I'm at right place, as I asked this question at stackoverflow.com where the question was closed and suggested I go ask here. I'll try to be more comprehensive also, looking at couple of replies ...
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Random assignment in blind experiments and “fair / just coin”

In blind experiments subjects are randomly assigned to one of groups. The most commonly used solution is to use (equivalent of) a coin toss, with the same probability to be assigned to each group. I ...
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106 views

If $x$ is a $p$-value, what do we call $x\cdot (1-x)$?

What do you call this function? $$f(x) = x(1 - x)$$ Does it have a name? It’s cropped up in at least two statistical formulas I've been asked to program so far. (In both cases, $x$ was something ...
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Motivation behind standard deviation?

Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) ...
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What is the probability of a coin landing tails 7 times in a row in a series of 150 coin flips?

If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability?
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306 views

Bar chart or histogram

What's the difference between a bar chart and a histogram?
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Why is median age a better statistic than mean age?

If you look at Wolfram Alpha or this Wikipedia page List of countries by median age Clearly median seems to be the statistic of choice when it comes to ages. I am not able to explain to myself ...
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Orthant probability / passage-time

I'm having a hard time solving this problem. Would appreciate any hint! (and special thanks to Byron Schmuland for answering my previous 2 questions. This 3rd question is different.) Let $e_t$: ...
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Sufficient statistics vs. Bayesian sufficient statistics

Given sample data $x_1, \ldots, x_n$ generated from a probability distribution $f(x|\theta)$ ($\theta$ being an unknown parameter), a statistic $T(x_1, \ldots, x_n)$ of the sample data is called ...
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What is the intuitive relationship between SVD and PCA

Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important ...
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easy to implement method to fit a power function (regression)

I want to fit to a dataset a power function ($y=Ax^B$). What is the best and easiest method to do this. I need the $A$ and $B$ parameters too. I'm using in general financial data in my project, which ...
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Instantaneous Z-Score Value

I had a question about an instantaneous z score. The question reads as $$P(z=2)$$ I think that this is 0, as z-score is the area under the curve and could be found by integrating the PDF over the ...
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Beta function derivation

How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments? I'm pretty sure this is ...
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Optimal sample size?

What's the simplest formula to calculate the optimal sample size from a population size? I've been reading a little about it and they ask me stuff like confidence level or confidence interval which I ...
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What is the expected number of runs of same color in a standard deck of cards?

Standard deck has $52$ cards, $26$ Red and $26$ Black. A run is a maximum contiguous block of cards, which has the same color. Eg. $(R,B,R,B,...,R,B)$ has $52$ runs. $(R,R,R,...,R,B,B,B,...,B)$ has ...
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probability and statistics: Does having little correlation imply independence?

Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables?
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How to calculate correlation between poll statistics and one's answers?

I have statistics of $100$ questions which can be answered either "yes" or "no": 1) $63.3 - 36.7$ ($63.3\%$ respondents answered "yes" and $36.7\%$ answered "no") 2) $30.1 - 69.9$ ... 100) $88.0 - ...
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Is there a simple test for uniform distributions?

I have a function that (more or less) is supposed to select a small number m of random number from the range [1,n] (for some n >> m) and I need to test that it work. Is there an easy to implement test ...
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Determine speed of the object at the current time by the non-uniform time sample

Here is a time sample: $Q = \{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\}$ and rules: (1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$ (2) $x_i$ comes with error: $x_i = ...
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247 views

What is the probability that exactly $i$ heads will result from $n$ coin tosses?

It seems like an extremely easy question, but I can't solve it. All I know is that it is a $\frac{1}{2^n}$ chance that all tosses will result in heads and $\sum_{k=0}^{n} f(k, n)$ is equal to 1 given ...
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How can I calculate this expected rate?

In DotA, there is a character called "Axe". Every time he is attacked, he has a chance to spin ($17\%$) his blade and deal damage based on what level the skill is, $100 / 125 / 150 / 175$ damage for ...
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How do I calculate expected value of partial normal distribution?

Suppose you have a normal distribution with mean=0, and stdev=1. So the expected value is 0. Now suppose you limit the outcomes, such that no values can be below 0. So 50% of values now equal 0, ...
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293 views

Uniqueness of Characterstic Functions in Probability

According to Wikipedia, a characteristic function completely determines the properties of a probability distribution. This means it must be unique. However, the definition give is: ...
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Why does Benford's Law (or Zipf's Law) hold?

Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural ...
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What can we conclude from correlation?

I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were ...
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How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula ...
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Real life usage of Benford's Law

I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.