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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How often does it happen that the oldest person alive dies?
Today, we are brought the sad news that Europe's oldest woman died. A little over a week ago the oldest person in the U.S. unfortunately died. Yesterday, the Netherlands' oldest man died peacefully. ...
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2answers
14k views
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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Why is the error function defined as it is?
$\newcommand{\erf}{\operatorname{erf}}$
This may be a very naïve question, but here goes.
The error function $\erf$ is defined by
$$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$
Of ...
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6answers
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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 ...
18
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8answers
990 views
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.
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1answer
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StarCraft II: Ladder math
At the Blizzcon 2010, StarCraft II multiplayer panel, this stuff was supposed to explain the ladder matchmaking system. I look at this and go eh? what!?
Is any of this real? or are they just ...
18
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4answers
475 views
Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$
Does anyone have an example of two dependent random variables, that satisfy this relation?
$E[f(X)f(Y)]=E[f(X)]E[f(Y)]$
for every function $f(t)$.
Thanks.
*edit: I still couldn't find an example. I ...
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5answers
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Why do we use a Least Squares fit?
I've been wondering for a while now if there's any deep mathematical or statistical significance to finding the line that minimizes the square of the errors between the line and the data points.
If ...
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4answers
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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) ...
14
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0answers
967 views
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 ...
13
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3answers
874 views
How to intuitively understand eigenvalue and eigenvector?
I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman.
Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to ...
12
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5answers
740 views
Why “bother” with a null hypothesis at all?
(note: this is a very basic probability question, so it is highly probable (heh) that it is a duplicate)
Every time I am trying to get into statistics (again), I am always lost at hypothesis testing.
...
12
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3answers
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Variance of sample variance?
What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.
I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$
I know that ...
12
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4answers
418 views
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?
12
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3answers
676 views
Odds of guessing suit from a deck of cards, with perfect memory
While teaching my daughter why drawing to an inside straight is almost always a bad idea, we stumbled upon what I think is a far more difficult problem:
You have a standard 52-card deck with 4 suits ...
11
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5answers
8k views
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?
11
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3answers
274 views
Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
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4answers
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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 ...
11
votes
2answers
542 views
You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon?
This question was asked on a forum and while many argued that the answer is 14 (since the probability of you seeing bus 14 is maximum in this case), I argued against it that they were working ...
11
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1answer
657 views
Is there a proof of Benford's Law?
As stated by Wikipedia (here):
Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed ...
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2answers
1k views
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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3answers
617 views
With $n$ balls and $n$ bins, what is the probability that exactly $k$ bins have exactly $1$ ball?
I've got a balls and bins problem. Suppose I throw $n$ balls uniformly at random into $n$ bins. What is the probability that exactly $k$ bins end up with exactly $1$ ball?
I know this seems a ...
10
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2answers
186 views
What is the difference between probability and statistics?
Is it that probability is top-down (going from pure distributions to predictions about events) and statistics is bottom-up (going from specific events to predicting pure distributions?)
I'm pretty ...
10
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3answers
146 views
Finding a more direct way to reach $\mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) = \sigma^2$
Let $X_i$ be independent random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance ...
10
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1answer
216 views
metric in the Wasserstein space of gaussian measures
I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used.
In particular, I am ...
9
votes
4answers
793 views
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 ...
9
votes
1answer
172 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 ...
9
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0answers
221 views
Hyper Birthday Paradox?
There are $N$ buckets.
Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets.
At $t=1$, we expect that at least one bucket ...
9
votes
2answers
258 views
Fast computation/estimation of the nuclear norm of a matrix
The nuclear norm of a matrix is defined as the sum of its singular values, as given by the Singular Value Decomposition of the matrix itself. It is of central importance in Signal Processing and ...
8
votes
2answers
251 views
Why is polynomial regression considered a kind of linear regression?
Why is polynomial regression considered a kind of linear regression?
This is what I mean by polynomial regression. For example, the hypothesis function is
$$h(x; t_0, t_1, t_2) = t_0 + t_1 x + t_2 ...
8
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3answers
563 views
Intuition about the Central Limit Theorem
I'm studying statistics, and would like to better understand the Central Limit Theorem. The proof I found on Wikipedia requires some previous knowledge I do not currently possess.
Is there a quick ...
8
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2answers
357 views
What is the relationship between the Boltzmann distribution and information theory?
I'm reading a paper on Boltzmann machines (a type of neural network in Machine Learning), and it mentions that "The Boltzmann distribution has some beautiful mathematical properties and it is ...
8
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1answer
327 views
Monotonic behavior of a function
I have the following problem related to a statistics question:
Prove that the function defined for $x\ge 1, y\ge 1$,
...
8
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3answers
292 views
How would I figure out how many anagrams of mississippi don't contain the word psi?
I'm really confused how I'd calculate this. I know it's the number of permutations of mississippi minus the number of permutations that contain psi, but considering there's repetitions of those ...
7
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2answers
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Finding $E\left[\frac{\sum_{i=1}^n X_i^2}{(\sum_{i=1}^n X_i)^2}\right]$ of a sample of gamma random variables
Suppose $X_1,\ldots,X_n$ is a random sample from the $\Gamma(k,\lambda)$ distribution where $\lambda$
is unknown and $k$ is a positive integer and known. How can I find $$E\left[\frac{\sum_{i=1}^n ...
7
votes
3answers
811 views
What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6?
Similar to:
"What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?"
but we allow repeats so 1,1,2,2,3,4,4,4,4,5,5,6 would count.
My answer (or simulation) is flawed as I ...
7
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3answers
147 views
How come in statistics there is very little justification for the formulas used and proofs are almost nonexistent [closed]
I don't understand why people accept certain formulas in statistics without a mathematical proof style argument. You see this a lot in statistics textbooks and unfortunately this spills over with the ...
7
votes
1answer
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Sample Standard Deviation vs. Population Standard Deviation
I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values:
Sample
Population
I ...
7
votes
4answers
1k views
Intuition behind using complementary CDF to compute expectation for nonnegative random variables
I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here) and understand its mechanics, but I'm having trouble understanding the intuition behind ...
7
votes
4answers
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What is the deepest / most interesting known connection between Trigonometry and Statistics?
I'm teaching both at the same time to different classes in high school, so I just wondered about this.
Added by OP on 16.May.2011 (Beijing time)
I mean Statistics only, without Probability. In ...
7
votes
2answers
550 views
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 ...
7
votes
4answers
2k views
Consecutive Coin Toss with static tosses
I'm writing an algorithm for a coin toss problem. But I have a problem understanding the calculation given.
Here is the question:
You have an unbiased coin which you want to keep tossing until ...
7
votes
3answers
143 views
Stein's lemma condition
(Apologies if I break some conventions, this is my first time posting!)
I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable ...
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2answers
220 views
Mean and Median in a Classic River Crossing Problem
Consider the following classic problem:
Four people on the west side of a river wish to use their single boat to get to the east side of a river. Each boat ride can hold at most two people, and the ...
7
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2answers
267 views
In what sense is the Jeffreys prior invariant?
I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. Most texts I've read online make some comment to the effect that the Jeffreys prior is ...
7
votes
1answer
190 views
Tuning the birthday paradox
I have limited access to a collection $X_1,\ldots,X_m$ of sets of positive integers. Each $X_i$ is "moderately large" (a brief survey has found them to contain about $10^6$ elements in each set), but ...
7
votes
2answers
398 views
Buckets of Balls, Will one fill if I add another Ball?
I was refereed here by stackoverflow.com. With some searching I found this: another balls and bins question, but its not quite what I am looking for. Rather the inverse. IE the expected number of ...
7
votes
3answers
590 views
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 ...
6
votes
5answers
507 views
Coin tosses until I'm out of money
The question I think is a simple one, but I've been unable to answer or find an answer for it yet:
There's a simple game: if you flip heads you win a dollar (from the house), but if you flip tails ...
6
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1answer
267 views
M.SE reputation distribution
What distribution does the reputation points per user follow on math.SE (or on entire stackexchange)?
Is there a mathematical explanation/model of it?
