A continuous random variable X follows a normal distribution, denoted as $X \sim \mathcal{N}(\mu,\,\sigma^{2})$. The normal distribution is characterized by its bell shape and symmetry. The majority of the values are concentrated around the mean, and there are no extreme values. It can be viewed as ...
A continuous random variable X follows a log-normal distribution if its natural logarithm is normally distributed. The log-normal distribution is useful in modeling continuous random variables that are constrained to be positive. It is denoted as $X \sim \text{LogNormal}(\mu, \sigma^2)$, where $\mu...
The Central Limit Theorem (CLT) is a fundamental concept in statistics, explaining why the distribution of sample means approximates a normal distribution, often known as the bell curve, as the sample size becomes larger, irrespective of the population's original distribution...
A normal distribution (often referred to as the normal curve or Gaussian distribution) is a continuous probability distribution that is symmetric about the mean, where most of the observations cluster around the central peak and taper off symmetrically towards both ends. Many real-world datasets suc...
Multiple linear regression is a statistical technique used to model the relationship between a single dependent variable and two or more independent variables. It extends the concept of simple linear regression by incorporating multiple predictors to explain the variability in the dependent variable...
Covariance is a fundamental statistical measure that quantifies the degree to which two random variables change together. It indicates the direction of the linear relationship between variables...
Simple linear regression is a fundamental statistical method used to model the relationship between a single dependent variable and one independent variable. It aims to find the best-fitting straight line through the data points, which can be used to predict the dependent variable based on the indep...
Statistics is an empirical science, focusing on data-driven insights for real-world applications. This guide offers a concise exploration of statistical fundamentals, aimed at providing practical knowledge for data analysis and interpretation...
Bayesian and frequentist statistics are two distinct approaches to statistical inference. Both approaches aim to make inferences about an underlying population based on sample data. However, the way they interpret probability and handle uncertainty is fundamentally different...
Probability trees are a visual representation of all possible outcomes of a probabilistic experiment and the paths leading to these outcomes. They are especially helpful in understanding sequences of events, particularly when these events are conditional on previous outcomes...
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In the realm of computer science, 'sorting' refers to the process of arranging a collection of items in a specific, predetermined order. This order is based on certain criteria that are defined beforehand...
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G艂贸wnym celem tego kursu jest zapoznanie uczestnik贸w z j臋zykiem programowania Python - od podstaw po bardziej zaawansowane zagadnienia. Kurs zosta艂 zaprojektowany tak, aby uczestnik m贸g艂 p艂ynnie przechodzi膰 przez kolejne etapy nauki, jednocze艣nie zdobywaj膮c praktyczne umiej臋tno艣ci...
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Evaluation metrics are essential tools for assessing the performance of statistical and machine learning models. They provide quantitative measures that help us understand how well a model is performing and where improvements can be made. In both classification and regression tasks, selecting approp...
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Anomaly detection involves identifying data points that significantly differ from the majority of the data, often signaling unusual or suspicious activities. This technique is widely used across various domains, such as fraud detection, manufacturing, and system monitoring...
Autocovariance functions describe how values of a time series relate to their lagged counterparts, measuring the joint variability between a series at time $t$ and its value at a previous time $t-k$ (where $k$ is the lag). In autoregressive models, these relationships are expressed through coefficie...
A sequence is an ordered list of numbers that can be viewed as a function mapping each natural number $n$ to a specific value $a_n$. More formally, a sequence ${a_n}$ is a function whose domain is the set of natural numbers, and the values are called the terms of the sequence...
A difference equation (also known as a recurrence relation) defines each term of a sequence based on previous terms. In some cases, the general term of a sequence is given explicitly (e.g., $a_n = 3n + 2$, resulting in the sequence $5, 8, 11, \dots$). However, more commonly, a difference equation pr...
The Yule-Walker equations are a set of linear equations that relate the autocorrelations of an autoregressive (AR) process to its parameters. These equations are crucial for estimating the parameters of AR models and for understanding the autocorrelation structure of the process...
The random walk is a fundamental and widely used time series model, often applied in finance to represent stock prices and other economic indicators. The idea behind the random walk is that the value of the process at time $t$ is the sum of its value at time $t-1$ and a random shock (or noise). Esse...
The backward shift operator (denoted by $B$) is a powerful tool in time series analysis, used to simplify the notation and manipulation of time series models. The operator shifts the time index of a time series back by one period, making it useful in autoregressive, moving average, and mixed models...
In time series modeling, invertibility is the property of a model that allows the innovation process (also called the noise or disturbance process) to be expressed as a function of the observed series and its past values. This is particularly relevant for Moving Average (MA) models...
Logistic regression is a statistical method used for modeling the probability of a binary outcome based on one or more predictor variables. It is widely used in various fields such as medicine, social sciences, and machine learning for classification problems where the dependent variable is dichotom...
Correlation is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is a fundamental concept in statistics, enabling researchers and analysts to understand how one variable may predict or relate to another. The most commonly used corre...
Conditional Probability is the likelihood of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, representing the probability of event $A$ happening, assuming event $B$ has already taken place. This concept is crucial in understanding dependent events in prob...