Notes in Math for RAI 2

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Published	10/15/2024	{{c1::Numerical analysis}} involves developing, analyzing, and implementing algorithms to obtain approximate solutions to complex mathematical problem…
Published	10/15/2024	In numerical analysis, errors can be broadly classified into {{c1::truncation errors}} and {{c2::round-off errors}}.
Published	10/15/2024	A {{c1::truncation error}} occurs when an infinite process is approximated by a finite one, such as using a finite number of terms in a series.
Published	10/15/2024	{{c1::Round-off error}} arises when a number is approximated due to the limited number of significant digits available in the representation.
Published	10/15/2024	{{c1::Significant figures}} are the digits in a number that carry meaningful information about its precision.
Published	10/15/2024	The {{c1::accuracy}} of a measurement refers to how close it is to the true value, while {{c2::precision}} refers to how close repeated measurements a…
Published	10/15/2024	The {{c1::Taylor Series}} is a mathematical series used to approximate functions by expressing them as an infinite sum of terms calculated from the va…
Published	10/15/2024	The {{c1::remainder}} or error term in a Taylor Series provides an estimate of the difference between the true value of the function and the approxima…
Published	10/15/2024	In numerical algorithms, the error that occurs due to approximating a function using a finite number of terms in the Taylor Series is called the {{c1:…
Published	10/15/2024	{{c1::Floating-point representation}} is a method of representing real numbers in a way that can support a wide range of values by using a fixed numbe…
Published	10/15/2024	{{c1::Round-off errors}} occur when numbers having limited significant figures are used to represent exact numbers.
Published	10/15/2024	{{c1::Significant figures}} are the digits in a number that can be used with confidence, consisting of the number of certain digits plus one estimated…
Published	10/15/2024	{{c1::Truncation errors}} arise when approximations are used to represent exact mathematical procedures.
Published	10/15/2024	The {{c1::true error}} is defined as the difference between the true value and the approximation.
Published	10/15/2024	The {{c1::true fractional relative error}} is calculated by dividing the true error by the true value, often expressed as a percentage.
Published	10/15/2024	In an {{c1::iterative algorithm}}, the current approximation is calculated using the previous approximation, and the process repeats until a desired a…
Published	10/15/2024	The {{c1::Taylor Series}} provides a way to approximate a function by expanding it into an infinite sum of terms based on the derivatives of the funct…
Published	10/15/2024	In floating-point representation, a number is expressed as {{c1::\( m \times b^e \)}}, where \( m \) is the mantissa, \( b \) is the base, and \( e \)…
Published	10/15/2024	{{c1::Numerical differentiation}} is the process of estimating the derivative of a function using discrete data points.
Published	10/15/2024	The {{c1::Taylor Series}} is used to approximate a function and its derivatives, providing a basis for numerical differentiation and error analysis.
Published	10/15/2024	{{c1::Error propagation}} refers to how errors in the input values propagate through a mathematical process to affect the output.
Published	10/15/2024	The {{c1::condition number}} measures the sensitivity of a function’s output to changes in its input, indicating the stability of a numerical computat…
Published	10/15/2024	A function with a high {{c1::condition number}} is said to be ill-conditioned, meaning small changes in the input can cause large changes in the outpu…
Published	10/15/2024	The {{c1::central difference}} method improves the accuracy of numerical differentiation by averaging the forward and backward differences.
Published	10/15/2024	{{c1::Truncation error}} in numerical differentiation arises from ignoring higher-order terms in the Taylor Series expansion.
Published	10/15/2024	In numerical differentiation, the {{c1::step size}} plays a crucial role in balancing the trade-off between truncation error and round-off error.
Published	10/15/2024	{{c1::Optimisation}} is the process of making the best or most effective use of resources, minimizing or maximizing a certain quantity.
Published	10/15/2024	A {{c1::local minimum}} of a function is a point where the function value is lower than at all nearby points, but not necessarily the lowest overall v…
Published	10/15/2024	In mathematical terms, {{c1::convexity}} refers to a function where a line segment joining any two points on its graph lies above or on the graph.
Published	10/15/2024	The {{c1::Hessian matrix}} is used in optimization to determine whether a critical point is a local maximum, minimum, or a saddle point.
Published	10/15/2024	In optimization problems, {{c1::constraints}} are conditions that the solution must satisfy, such as limits on resources.
Published	10/15/2024	A {{c1::global minimum}} is the lowest point of a function over its entire domain, not just within a neighborhood of points.
Published	10/15/2024	The {{c1::gradient}} of a function is a vector that points in the direction of the greatest rate of increase of the function.
Published	10/15/2024	The {{c1::Lagrange multiplier method}} is used to find the local maxima and minima of a function subject to equality constraints.
Published	10/15/2024	{{c1::Gauss’s method of least squares}} minimizes the sum of the squares of the differences between observed and predicted values.
Published	10/15/2024	The {{c1::steepest ascent method}} is an optimization technique that involves moving in the direction of the gradient to find a local maximum.
Published	10/15/2024	In {{c1::linear programming}}, the goal is to optimize a linear objective function subject to linear equality and inequality constraints.
Published	10/15/2024	{{c1::Slack variables}} are introduced in linear programming to convert inequality constraints into equality constraints.
Published	10/15/2024	The {{c1::Pascal's identity}} states that \( \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k} \).
Published	10/15/2024	The sum of an infinite geometric series with a common ratio \( r \) such that \( 0 < r < 1 \) is given by {{c1::\( \frac{1}{1-r} \)}}.
Published	10/15/2024	The {{c1::binomial theorem}} describes the algebraic expansion of powers of a binomial. It is written as \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^…
Published	10/15/2024	The {{c1::permutation}} of a set of items is an arrangement of those items in a particular order, calculated as \( P(n, k) = \frac{n!}{(n-k)!} \).
Published	10/15/2024	A {{c1::combination}} is a selection of items from a larger pool where the order does not matter, calculated as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} …
Published	10/15/2024	Optimization involves choosing a variable that {{c1::minimizes}} or {{c2::maximizes}} a certain quantity of interest, possibly under some constraints.
Published	10/15/2024	A set \( C \) is {{c1::convex}} if for any two points \( x_1, x_2 \in C \), the line segment connecting them is also within \( C \).
Published	10/15/2024	The key difference between a {{c1::global minimizer}} and a {{c2::local minimizer}} is that the former achieves the absolute minimum over the entire d…
Published	10/15/2024	A function \( f(x) \) is convex if its {{c1::Hessian matrix \( H \)}} is {{c2::positive semi-definite}}.
Published	10/15/2024	The gradient \( \nabla f(x) \) represents the {{c1::steepest ascent}} direction of the function \( f(x) \).
Published	10/15/2024	In a convex optimization problem, a {{c1::local minimizer}} is also a {{c2::global minimizer}}.
Published	10/15/2024	The {{c1::steepest ascent method}} is an iterative optimization technique that involves moving in the direction of the {{c2::gradient}} to find a loca…
Published	10/15/2024	The {{c1::Hessian matrix}} is used in optimization to determine whether a point is a {{c2::local minimum}} or a {{c3::local maximum}}.
Published	10/15/2024	In the context of optimization, {{c1::unconstrained optimization}} refers to problems that do not have {{c2::constraints}} on the variables.
Published	10/15/2024	The method of {{c1::random search}} is an optimization technique where the solution space is explored by randomly sampling points.
Published	10/15/2024	The {{c1::univariate search}} method improves the approximation by changing {{c2::one variable at a time}} while keeping the others constant.
Published	10/15/2024	{{c1::Gradient methods}} in optimization use the {{c2::gradient}} of the function to find the direction of steepest ascent or descent.
Published	10/15/2024	The {{c1::steepest ascent}} method involves finding both the {{c2::direction}} and the {{c3::step size}} for moving towards the maximum.
Published	10/15/2024	The {{c1::step size \( h \)}} in the steepest ascent method is crucial for determining the {{c2::optimal movement}} along the gradient direction.
Published	10/15/2024	{{c1::Convex functions}} and {{c2::convex sets}} are fundamental in ensuring that local minima are also global minima in convex optimization problems.
Published	10/15/2024	A {{c1::geometric series}} is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called …
Published	10/15/2024	The sum of an infinite geometric series is given by {{c1::\(\frac{1}{1-r}\)}} for \( 0 < r < 1 \).
Published	10/15/2024	The {{c1::binomial theorem}} provides a formula for expanding expressions that are raised to any positive integer power.
Published	10/15/2024	{{c1::Set theory}} is a branch of mathematical logic that studies sets, which are collections of objects.
Published	10/15/2024	A {{c1::subset}} is a set where every element in the subset is also in the larger set. If set B is a subset of set A, it is denoted as \( B \subseteq …
Published	10/15/2024	The {{c1::union}} of two sets contains all the elements that are in either set or in both sets, denoted as \( A \cup B \).
Published	10/15/2024	The {{c1::intersection}} of two sets contains all the elements that are in both sets, denoted as \( A \cap B \).
Published	10/15/2024	The {{c1::complement}} of a set contains all the elements that are in the universal set but not in the given set, denoted as \( A' \).
Published	10/15/2024	{{c1::Probability space}} is defined by three elements: a {{c2::sample space}} \( \Omega \), an {{c3::event space}} \( \mathcal{F} \), and a {{c4::pro…
Published	10/15/2024	{{c1::Conditional probability}} is the probability of an event occurring given that another event has already occurred, represented as \( P(A\|B) = \fr…
Published	10/15/2024	Two events are said to be {{c1::independent}} if the occurrence of one does not affect the probability of the other, mathematically represented as \( …
Published	10/15/2024	{{c1::Bayes' theorem}} relates the conditional probability of two events and is used to update the probability of a hypothesis based on new evidence.
Published	10/15/2024	A {{c1::random variable}} is a function that maps outcomes from a sample space to the real number line.
Published	10/15/2024	The {{c1::probability mass function (PMF)}} gives the probability that a discrete random variable takes on a specific value.
Published	10/15/2024	The {{c1::cumulative distribution function (CDF)}} of a discrete random variable is a function that gives the probability that the variable takes a va…
Published	10/15/2024	The {{c1::expectation}} of a random variable is the weighted average of all possible values that the random variable can take, weighted by their proba…
Published	10/15/2024	A {{c1::Bernoulli random variable}} takes the value 1 with probability \( p \) and 0 with probability \( 1 - p \).
Published	10/15/2024	The {{c1::binomial distribution}} models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of …
Published	10/15/2024	A {{c1::geometric random variable}} models the number of Bernoulli trials needed to get the first success.
Published	10/15/2024	The {{c1::Poisson distribution}} gives the probability of a given number of events occurring in a fixed interval of time or space, assuming they occur…
Published	10/15/2024	The {{c1::variance}} of a random variable is a measure of how spread out its values are around the mean.
Published	10/15/2024	The {{c1::standard deviation}} is the square root of the variance and provides a measure of the spread of the random variable in the same units as the…
Published	10/15/2024	The {{c1::probability density function (PDF)}} is the continuous version of the PMF and describes the likelihood of a continuous random variable takin…
Published	10/15/2024	The {{c1::cumulative distribution function (CDF)}} for a continuous random variable is the probability that the variable takes a value less than or eq…
Published	10/15/2024	The {{c1::expectation}} of a continuous random variable is calculated by integrating the product of the variable’s value and its PDF over the possible…
Published	10/15/2024	The {{c1::variance}} of a continuous random variable is the expectation of the squared deviation of the variable from its mean.
Published	10/15/2024	A {{c1::uniform random variable}} is a continuous random variable that has equal probability for all values in a given interval.
Published	10/15/2024	An {{c1::exponential random variable}} is a continuous random variable whose PDF is given by \( f_X(x) = \lambda e^{-\lambda x} \) for \( x \geq 0 \),…
Published	10/15/2024	The {{c1::mean}} of a random variable is a measure of its central tendency, while the {{c2::median}} is the point at which half of the distribution li…
Published	10/15/2024	The {{c1::mode}} is the value of a random variable at which the PDF attains its maximum.
Published	10/15/2024	The {{c1::Gaussian distribution}} (or normal distribution) has a PDF given by \( f_X(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2 \si…
Published	10/15/2024	The {{c1::Central Limit Theorem}} states that the sum of a large number of independent random variables, regardless of their original distribution, te…
Published	10/15/2024	A {{c1::joint probability distribution}} describes the probability of two or more random variables occurring simultaneously.
Published	10/15/2024	The {{c1::joint probability mass function (PMF)}} is the probability that two or more discrete random variables take on specific values.
Published	10/15/2024	The {{c1::marginal distribution}} of a random variable is obtained by summing or integrating the {{c2::joint distribution}} over the other variables.
Published	10/15/2024	Two random variables are {{c1::independent}} if the joint distribution equals the product of their marginal distributions.
Published	10/15/2024	The {{c1::Cauchy-Schwarz inequality}} is a mathematical inequality used in statistics and probability, which states that for any random variables \( X…
Published	10/15/2024	The {{c1::covariance}} between two random variables measures how much they change together, and is defined as \( \text{Cov}(X, Y) = E[(X - E[X])(Y - E…
Published	10/15/2024	The {{c1::correlation coefficient}} \( \rho \) is a normalized measure of the linear relationship between two random variables, taking values between …
Published	10/15/2024	{{c1::Uncorrelated}} variables have a correlation coefficient of 0, but this does not necessarily imply that the variables are independent.
Published	10/15/2024	The {{c1::law of total expectation}} states that the {{c2::expected value}} of a random variable is the {{c3::weighted average of its conditional expe…
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