# Questions tagged [mathematical-statistics]

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

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8answers
10k views

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

What do they mean when they say "random variable"?
11answers
54k views

### 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.
9answers
82k 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 ...
3answers
6k 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 ...
5answers
13k 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 ...
14answers
63k 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'...
3answers
6k views

9answers
83k 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 ...
5answers
33k 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 "...
2answers
30k views

### What is the distribution of the sum of non i.i.d. gaussian variates?

If $X$ is distributed $N(\mu_X, \sigma^2_X)$, $Y$ is distributed $N(\mu_Y, \sigma^2_Y)$ and $Z = X + Y$, I know that $Z$ is distributed $N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$ if X and Y are ...
4answers
3k views

### Good resources (online or book) on the mathematical foundations of statistics

Before I ask my question, let me give you a bit of background about what I know about statistics so that you have a better sense of the types of resources that I'm looking for. I'm a graduate student ...
3answers
16k views

### Asymptotic distribution of sample variance of non-normal sample

This is a more general treatment of the issue posed by this question. After deriving the asymptotic distribution of the sample variance, we can apply the Delta method to arrive at the corresponding ...
1answer
935 views

### Estimating parameters for a binomial

First of all I'd like to precise that I'm not an expert of the subject. Suppose to have two random variables $X$ and $Y$ that are binomial, respectively $X\sim B(n_1,p)$ and $Y\sim B(n_2,p),$ note ...
1answer
4k views

### Moment generating function of the inner product of two gaussian random vectors

Can anybody please suggest how I can compute the moment generating function of the inner product of two Gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each other?...
3answers
13k 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 ...
2answers
3k views

### What is the intuition behind defining completeness in a statistic as being impossible to form an unbiased estimator of $0$ from it?

In classical statistics, there is a definition that a statistic $T$ of a set of data $y_1, \ldots, y_n$ is defined to be complete for a parameter $\theta$ it is impossible to form an unbiased ...
2answers
6k views

### What is the distribution of $R^2$ in linear regression under the null hypothesis? Why is its mode not at zero when $k>3$?

What is the distribution of the coefficient of determination, or R squared, $R^2$, in linear univariate multiple regression under the null hypothesis $H_0:\beta=0$? How does it depend on the number ...
3answers
5k views

### Tool for generating correlated data sets

Does anyone know of a tool that I can use to generate a set of data with known correlations (and to put the icing on the cake - output this in json,csv,txt or some common format)? I am working on ...
1answer
1k views

### What is meant by using a probability distribution to model the output data for a regression problem?

Often a theoretical text will say something like, 'a probability distribution may be used to model the data' or, 'assume a probability distribution such as normal or Lognormal for the outputs'. ...
2answers
5k views

### Transformation Chi-squared to Normal distribution

The relationship between the standard normal and the chi-squared distributions is well known. I was wondering though, is there a transformation that can lead from a $\chi^2 (1)$ back to a standard ...
12answers
10k views

### 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 ...
1answer
38k 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 ...
3answers
3k views