The mean of a random variable is also known as its expected value. It is a measure of the central tendency of the distribution of the random variable. The expected value of a discrete random variable X with possible values x1, x2, …, xn and corresponding probabilities P(X = x1), P(X = x2), …, P(X…
Standard deviation and variance are measures of variability that provide information about how spread out a dataset is. The standard deviation is the square root of the variance, and it measures the average deviation of each data point from the mean. In general, the larger the standard deviation or variance, the more spread out the…
Mean deviation is a measure of variability that describes the average difference between the values in a dataset and their mean. It is also known as mean absolute deviation (MAD). The formula for calculating the mean deviation is: Mean Deviation = (Σ |xi – x̄|) / n where: Σ represents the sum of the absolute…
In probability and statistics, the mean (also known as the average) is a measure of central tendency. It is calculated by summing up all the values in a data set and dividing by the number of values. The formula for the mean is: mean = (sum of all values) / (number of values) For example,…
The total probability theorem is a fundamental concept in probability theory that allows us to calculate the probability of an event by conditioning on other events. The theorem states that if we have a set of events {A1, A2, A3, …, An} that are mutually exclusive and exhaustive, meaning that one and only one of…
In probability theory, two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event occurring. In other words, if the probability of A occurring is not affected by whether or not B occurs, and vice versa, then A and B are independent events.…
The addition rule of probability states that the probability of the occurrence of either of two mutually exclusive events is the sum of their individual probabilities. In other words, if A and B are two events that cannot happen simultaneously, then the probability of either A or B occurring is given by: P(A or B)…
The determinant of a square matrix of order up to three can be calculated as follows: For a 1×1 matrix: The determinant of a 1×1 matrix is simply the value of the only element in the matrix. For a 2×2 matrix: The determinant of a 2×2 matrix is calculated as follows: |a b| |c d|…
Matrices are mathematical objects consisting of rows and columns of numbers. They are commonly used in many fields of mathematics, physics, engineering, and computer science. One of the most important concepts in matrix theory is the idea of elementary row and column transformations. These are operations that can be performed on a matrix to transform…
In linear algebra, the transpose of a matrix is an operation that flips the matrix over its diagonal, reflecting its rows and columns. The transpose of a matrix A is denoted by A^T. To compute the transpose of a matrix, you simply write the rows of the matrix as columns, and the columns as rows.…
