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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113
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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 ...
81
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9answers
22k views

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. ...
36
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2answers
15k views

Is there a mathematical reason why chocolate chip cookies have 37% (1/e) chocolate in them?

Someone once briefly explained to me why it is that chocolate chip cookies have 37% chocolate in them. To the best of my memory it has to do with the way trying to place dots in a circle in a random ...
35
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5answers
16k 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 ...
31
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10answers
2k 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.
30
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3answers
4k views

Why does this not seem to be random?

I was running a procedure to be like one of those games were people try to guess a number between 1 and 100 where there are 100 people guessing.I then averaged how many different guesses there are. ...
30
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2answers
4k views

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 ...
26
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8answers
5k views

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 ...
25
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11answers
5k views

Two dice thrown, one comes up 6

If my friend throws two dice, and covers them up, but I see that one of them was a 6, what's the probability that they were both 6s given this knowledge? I'm under the impression that the answer is ...
25
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2answers
323 views

Statistics Primer for the Unwary Mathematician

I have a new position in a biology department (after being housed in a maths department) working on cognitive and population modeling. People in my lab are asking for help with applying statistical ...
23
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6answers
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 ...
22
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8answers
5k views

If a lottery has 300 tickets, shouldn't I win every 300 times I play

Suppose I play a lottery that has 300 tickets. I can only buy one ticket per draw. Statistically speaking, shouldn't I win once every 300 draws? Is it more complicated than this? Edit This question ...
22
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2answers
386 views

How long does it take a person with this “cheating” data-gathering strategy to achieve a desired result?

I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, I will cheat. Whenever I fail to have enough evidence, I will ...
20
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1answer
2k views

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 ...
20
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5answers
2k 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 ...
18
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2answers
7k views

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 ...
17
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3answers
3k views

1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?

A hypothetical example: You have a 1/1000 chance of being hit by a bus when crossing the street. However, if you perform the action of crossing the street 1000 times, then your chance of being ...
17
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7answers
21k 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?
17
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4answers
2k views

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) ...
17
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1answer
2k 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 ...
15
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2answers
2k 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 ...
15
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4answers
23k views

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 ...
15
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4answers
10k views

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 ...
15
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2answers
2k views

Average IQ of Mensa

I was wondering, what the average IQ at Mensa is. Mensa is a group of people with an IQ of at least 130. And the IQ is normally distribed with $\mu = 100$ and $\sigma = 15$. My idea was this: To ...
14
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3answers
4k views

Intuitive explanation of a definition of the Fisher information

I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this: $$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
14
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3answers
2k 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 ...
13
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5answers
2k 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 ...
13
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7answers
5k views

Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
13
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3answers
4k views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ ...
12
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5answers
830 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
votes
4answers
6k 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 ...
12
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4answers
635 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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2answers
359 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 ...
12
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3answers
458 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 ...
12
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1answer
344 views

Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from ...
11
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6answers
957 views

Why get the sum of squares instead of the sum of absolute values?

I'm self-studying machine learning and getting into the basics of linear regression models. From what I understand so far, a good regression model minimizes the sum of the squared differences between ...
11
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1answer
27k views

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 ...
11
votes
3answers
301 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 $p_1$ ...
11
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2answers
2k views

derivative of cost function for Logistic Regression

I am going over the lectures on Machine Learning at Coursera. I am struggling with the following. How can the partial derivative of ...
11
votes
2answers
701 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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3answers
308 views

How many books are in a library?

My cousin is at elementary school and every week is given a book by his teacher. He then reads it and returns it in time to get another one the next week. After a while we started noticing that he ...
11
votes
1answer
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 ...
11
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1answer
3k views

Is there a proof of Benford's Law? [duplicate]

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 ...
11
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0answers
209 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
10
votes
3answers
950 views

What is the Probability that a Knight stays on chessboard after N hops?

Say a $8 \times 8$ chessboard as per picture. A position is represented here by co-ordinates $(x,y)$. A move is aslo considered as valid, where the Knight lands outside the chessboard [ For eg. ...
10
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3answers
976 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
266 views

Does variance do any good to gambling game makers?

People always like to evaluate the variance, but is there any way for variance to be interesting to the gambling game makers? In another word, what is a pratical gambling game that involving some ...
10
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2answers
242 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
votes
2answers
976 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 ...
10
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3answers
261 views

How can you trust a bettor?

Two friends of mine are trying to convince me that they are good at betting. They infact have a certain profit after n bets. They provided me with all the bets they have made so far: odds, outcome and ...