MatStat#
Mathematical statistics is a branch of mathematics that provides a theoretical foundation for statistical methods. It focuses on developing and analyzing tools for collecting, analyzing, interpreting, and presenting data based on formal mathematical principles.
Events#
An event is the outcome of an observation. Mathematical statistics generally operates with events and has corresponding notation and a system of terms for them. This section describes them.
Event#
A lowercase letter, such as \(\omega\), usually denotes a simple event—an individual outcome of an experiment that occurs on its own.
An uppercase letter, such as \(E\), denotes a compound event—a set of simple events, for example, \(E = \left\{ \omega_1, \omega_2, \ldots, \omega_n \right\}\).
For instance, in a dice roll, the event that the result is 2 can be denoted as \(\omega = 2\), while the event that the result is an even number can be denoted as \(E = \left\{2, 4, 6\right\}\).
Operations#
\(\omega \in E\) implies that \(E\) occurs when \(\omega\) occurs.
\(\omega \notin E\) implies that \(E\) occurs when \(\omega\) occurs.
\(E \subset F\) implies that the occurrence of \(E\) implies the occurrence of \(F\).
\(E \cap F\) implies the event that both \(E\) and \(F\) occurs.
\(E \cup F\) implies the event that at least one of \(E\) or \(F\) occures.
\(\overline{E}\) is the event that \(E\) does not occur.
Terminology#
There is a set of commonty used in matematical statistics terms.
Sample space is the set that contains all possible outcomes of an experiment. It is usually denoted by \(\Omega\).
Null event is the event tha contains no otcomse. It is ususally denoted by \(\emptyset\).
Mutually exclusive are those that could not appear together, so \(A \cap B = \emptyset\).
Probability#
Probability is a function that maps events to real numbers in the interval [0, 1], reflecting the long-run relative frequency with which the event occurs in repeated independent trials under identical conditions.
So, \(P(A)\) is the probability that event A occurs.
There is a set of properties that probability must follow:
\(P(\emptyset) = 0\).
\(P(E) = 1 - P(E^c)\).
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
\(\text{if } A \subseteq B \text{ then } P(A) \leq P(B)\).
\(P(A \cup B) = 1 - P(\overline{A} \cap \overline{B})\).
\(P(A \cap \overline{B}) = P(A) - P(A \cap B)\).
\(P\left(\bigcup_{i=1}^{n} E_i\right) \leq \sum_{i=1}^{n} P(E_i)\).
\(P\left(\bigcup_{i=1}^{n} E_i\right) \geq \max_i P(E_i\))
Check more detailed consideration of some of this formulas in the Probability page.
Distributions#
There is a set of random value distributions that, due to their properties, are useful in some specific applications.
The following table lists the most common distributions used in practice.
Distribution |
Type |
Parameters |
Support |
Typical Use Cases |
|---|---|---|---|---|
Bernoulli |
Discrete |
\(p\) (probability of success) |
\(x \in \{0, 1\}\) |
Binary outcomes (e.g., coin toss) |
Binomial |
Discrete |
\(n\) (trials), \(p\) (success prob.) |
\(x \in \{0, \dots, n\}\) |
# of successes in fixed # of trials |
Geometric |
Discrete |
\(p\) (success prob.) |
\(x \in \{1, 2, \dots\}\) |
Trials until first success |
Poisson |
Discrete |
\(\lambda\) (rate) |
\(x \in \{0, 1, 2, \dots\}\) |
Count of events in fixed interval |
Uniform |
Continuous |
\(a\) (min), \(b\) (max) |
\(x \in [a, b]\) |
Equal probability over an interval |
Normal (Gaussian) |
Continuous |
\(\mu\) (mean), \(\sigma^2\) (variance) |
\(x \in \mathbb{R}\) |
Natural phenomena, errors, CLT |
Exponential |
Continuous |
\(\lambda\) (rate) |
\(x \in [0, \infty)\) |
Time between Poisson events |
Gamma |
Continuous |
\(\alpha\) (shape), \(\beta\) (rate) |
\(x \in [0, \infty)\) |
Waiting times, reliability analysis |
Beta |
Continuous |
\(\alpha\), \(\beta\) (shape params) |
\(x \in [0, 1]\) |
Probabilities, Bayesian inference |
Chi-squared |
Continuous |
\(k\) (degrees of freedom) |
\(x \in [0, \infty)\) |
Hypothesis testing, variance estimates |
Student’s t |
Continuous |
\(\nu\) (degrees of freedom) |
\(x \in \mathbb{R}\) |
Small-sample inference for means |
F-distribution |
Continuous |
\(d_1\), \(d_2\) (degrees of freedom) |
\(x \in [0, \infty)\) |
ANOVA, comparing variances |
Log-normal |
Continuous |
\(\mu\), \(\sigma\) (log-space params) |
\(x \in (0, \infty)\) |
Skewed data, multiplicative processes |
Statistical testing#
Statistical testing is a formal procedure used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population.
In practice, it involves:
Formulating hypotheses:
The null hypothesis (\(H_0\)): a default assumption (e.g., “no effect”, “no difference”).
The alternative hypothesis (\(H_1\) or \(H_a\)): the assertion you want to test (e.g., “there is an effect”).
Calculating a test statistic from the data that summarizes the difference between what is observed and what is expected under the null hypothesis.
Determining a p-value, which tells you the probability of obtaining a result at least as extreme as the observed one, assuming the null hypothesis is true.
Making a decision:
If the p-value is below a predefined threshold (commonly 0.05), reject the null hypothesis.
Otherwise, do not reject it.
Statistical testing is widely used in scientific research, quality control, medicine, and many other fields.
Check more on statistical testing on the corresponding page.
Central limit theorem#
Here’s the improved version of your text:
The formulation of the central limit theorem is:
Let’s take \(N\) independent values \(x_i\) drawn from the same distribution with expected value \(\mu\) and standard deviation \(\sigma\).
We denote their sum as:
\(S_N = \sum_{i=1}^N x_i\)
The central limit theorem states that \(S_N \sim \mathcal{N}(\mu N, \sigma^2 N)\) — that is, \(S_N\) follows a normal distribution with mean \(\mu N\) and variance \(\sigma^2 N\).
Check more details in the corresponding page.