# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### What's the difference between variance and standard deviation?

I was wondering what the difference between the variance and the standard deviation is. If you calculate the two values, it is clear that you get the standard deviation out of the variance, but what ...
90k views

### Bottom to top explanation of the Mahalanobis distance?

I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still ...
85k views

### What's the difference between probability and statistics?

What's the difference between probability and statistics, and why are they studied together?
88k views

### Numerical example to understand Expectation-Maximization

I am trying to get a good grasp on the EM algorithm, to be able to implement and use it. I spent a full day reading the theory and a paper where EM is used to track an aircraft using the position ...
77k views

### Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
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### Maximum Likelihood Estimation (MLE) in layman terms

Could anyone explain to me in detail about maximum likelihood estimation (MLE) in layman's terms? I would like to know the underlying concept before going into mathematical derivation or equation.
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### Who Are The Bayesians?

As one becomes interested in statistics, the dichotomy "Frequentist" vs. "Bayesian" soon becomes commonplace (and who hasn't read Nate Silver's The Signal and the Noise, anyway?). In talks and ...
68k views

### Simple algorithm for online outlier detection of a generic time series

I am working with a large amount of time series. These time series are basically network measurements coming every 10 minutes, and some of them are periodic (i.e. the bandwidth), while some other aren'...
29k views

### If mean is so sensitive, why use it in the first place?

It is a known fact that median is resistant to outliers. If that is the case, when and why would we use the mean in the first place? One thing I can think of perhaps is to understand the presence of ...
118k views

### Validation Error less than training error?

I found two questions here and here about this issue but there is no obvious answer or explanation yet.I enforce the same problem where the validation error is less than training error in my ...
13k views

### What is meant by a “random variable”?

What do they mean when they say "random variable"?
7k views

### Famous statistical wins and horror stories for teaching purposes

I am designing a one year program in data analysis with a local community college. The program aims to prepare students to handle basic tasks in data analysis, visualization and summarization, ...
39k views

### How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about "...
42k views

### How does the reparameterization trick for VAEs work and why is it important?

How does the reparameterization trick for variational autoencoders (VAE) work? Is there an intuitive and easy explanation without simplifying the underlying math? And why do we need the 'trick'?
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### Bayesians: slaves of the likelihood function?

In his book "All of Statistics", Prof. Larry Wasserman presents the following Example (11.10, page 188). Suppose that we have a density $f$ such that $f(x)=c\,g(x)$, where $g$ is a known (nonnegative, ...
29k views

### Mutual information versus correlation

Why and when we should use Mutual Information over statistical correlation measurements such as "Pearson", "spearman", or "Kendall's tau" ?
9k views

### Why would parametric statistics ever be preferred over nonparametric?

Can someone explain to me why would anyone choose a parametric over a nonparametric statistical method for hypothesis testing or regression analysis? In my mind, it's like going for rafting and ...
8k views

### What is the most surprising characterization of the Gaussian (normal) distribution?

A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: $$\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ or its characteristic function. As recalled in this ...
17k views

### Central limit theorem for sample medians

If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a ...
4k views

### What are the breakthroughs in Statistics of the past 15 years?

I still remember the Annals of Statistics paper on Boosting by Friedman-Hastie-Tibshirani, and the comments on that same issues by other authors (including Freund and Schapire). At that time, clearly ...
15k views

### What is so cool about de Finetti's representation theorem?

From Theory of Statistics by Mark J. Schervish (page 12): Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their ...
3k views

### Are we exaggerating importance of model assumption and evaluation in an era when analyses are often carried out by laymen

Bottom line, the more I learn about statistics, the less I trust published papers in my field; I simply believe that researchers are not doing their statistics well enough. I'm a layman, so to speak. ...
49k views

### KL divergence between two multivariate Gaussians

I'm having trouble deriving the KL divergence formula assuming two multivariate normal distributions. I've done the univariate case fairly easily. However, it's been quite a while since I took math ...
65k views

### Why is it that natural log changes are percentage changes? What is about logs that makes this so?

Can somebody explain how the properties of logs make it so you can do log linear regressions where the coefficients are interpreted as percentage changes?
8k views

### Mathematical Statistics Videos

A question previously sought recommendations for textbooks on mathematical statistics Does anyone know of any good online video lectures on mathematical statistics? The closest that I've found are: ...
8k views

### Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$?

I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the ...
12k views

### Motivation for Kolmogorov distance between distributions

There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are: the Kolmogorov distance: the sup-distance between the ...
7k views

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### Is a strong background in maths a total requisite for ML?

I'm starting to want to advance my own skillset and I've always been fascinated by machine learning. However, six years ago instead of pursuing this I decided to take a completely unrelated degree to ...
19k views

### Is it possible to prove a null hypothesis?

As the question states - Is it possible to prove the null hypothesis? From my (limited) understanding of hypothesis, the answer is no but I can't come up with a rigorous explanation for it. Does the ...
2k views

### Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper ...
3k views

### How can I analytically prove that randomly dividing an amount results in an exponential distribution (of e.g. income and wealth)?

In this current article in SCIENCE the following is being proposed: Suppose you randomly divide 500 million in income among 10,000 people. There's only one way to give everyone an equal, 50,000 ...
42k views

### Things to consider about masters programs in statistics

It is admission season for graduate schools. I (and many students like me) am now trying to decide which statistics program to pick. What are some things those of you who work with statistics suggest ...
11k views

### Differences between Bhattacharyya distance and KL divergence

I'm looking for an intuitive explanation for the following questions: In statistics and information theory, what's the difference between Bhattacharyya distance and KL divergence, as measures of the ...
11k views

### Are there any examples of where the central limit theorem does not hold?

Wikipedia says - In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends ...
4k views

### Does a sample version of the one-sided Chebyshev inequality exist?

I am interested in the following one-sided Cantelli's version of the Chebyshev inequality: $$\mathbb P(X - \mathbb E (X) \geq t) \leq \frac{\mathrm{Var}(X)}{\mathrm{Var}(X) + t^2} \,.$$ Basically, ...
75k views

### Looking for a good and complete probability and statistics book

I never had the opportunity to visit a stats course from a math faculty. I am looking for a probability theory and statistics book that is complete and self-sufficient. By complete I mean that it ...
4k views

### I know the 95% confidence interval for ln(x), do I also know the 95% confidence interval of x?

Suppose the 95% confidence interval for $\ln(x)$ is $[l,u]$. Is it true that the 95% CI for $x$ is simply $[e^l, e^u]$? I have the intuition the answer is yes, because $\ln$ is a continuous function. ...
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If $\mathbf{x}$ and $\mathbf{y}$ are two independent random unit vectors in $\mathbb{R}^D$ (uniformly distributed on a unit sphere), what is the distribution of their scalar product (dot product) $\... 6answers 5k views ### If 'correlation doesn't imply causation', then if I find a statistically significant correlation, how can I prove the causality? I understand that correlation is not causation. Suppose we get high correlation between two variables. How do you check if this correlation is actually because of causation? Or,under what conditions, ... 3answers 3k views ### How to rigorously define the likelihood? The likelihood could be defined by several ways, for instance : the function$L$from$\Theta\times{\cal X}$which maps$(\theta,x)$to$L(\theta \mid x)$i.e.$L:\Theta\times{\cal X} \rightarrow \...
The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...