Mathematics and Probability and Statistics Flashcards
Refresh concepts for the FE Mechanical Exam
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What is the definition of a conic section in analytic geometry?
A conic section is a curve obtained by intersecting a cone with a plane, including circles, ellipses, parabolas, and hyperbolas.
How does differential calculus relate to rate of change?
Differential calculus studies the rates at which quantities change, primarily using derivatives to determine these rates.
Define ‘integral calculus’.
Integral calculus involves the calculation of the area under a curve to find integral values, which represent accumulations of quantities.
Explain ‘single-variable calculus’ in terms of limits.
Single-variable calculus involves functions of one variable and is grounded in the concept of limits, which approach a specific value.
What is ‘multivariable calculus’ used for?
Multivariable calculus extends calculus to functions with more than one variable, used in fields like engineering and economics for optimizing solutions.
Describe a ‘homogeneous ordinary differential equation’.
A homogeneous ordinary differential equation is one where all terms are a function of the dependent variable and its derivatives only.
What are the applications of Laplace transforms in solving differential equations?
Laplace transforms simplify the solving of differential equations by converting them from time domain to frequency domain.
How is matrix multiplication performed in linear algebra?
Matrix multiplication involves taking the dot product of rows of the first matrix with columns of the second matrix.
Explain ‘vector analysis’ in the context of physics.
Vector analysis deals with quantities having both magnitude and direction, useful in physics for describing forces and velocities.
What is ‘numerical approximation’ and its significance?
Numerical approximation involves estimating values for calculations that cannot be solved analytically, crucial in modeling real-world scenarios where precise solutions are impossible.
Explain the ‘error propagation’ in numerical methods.
Error propagation refers to how approximation errors in numerical methods can accumulate and affect the overall outcome of calculations.
What is the use of Taylor’s series in numerical methods?
Taylor’s series is used to approximate complex functions by a sum of its derivatives at a specific point, simplifying calculations.
Describe Newton’s method for finding roots.
Newton’s method is an iterative numerical technique for finding successively better approximations to the roots (or zeroes) of a real-valued function.
Define ‘normal distribution’ in statistics.
The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence.
What is a ‘binomial distribution’?
A binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent yes/no experiments.
Describe the concept of ‘expected value’ in probability.
Expected value is the weighted average of all possible values that a random variable can take on, each value weighted by its probability of occurrence.
What is ‘linear regression’ used for in statistics?
Linear regression is used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data.
Explain the significance of ‘standard deviation’ in data analysis.
Standard deviation measures the amount of variation or dispersion in a set of numeric values, indicating how much the values differ from the mean.
How is ‘confidence interval’ calculated and used?
A confidence interval is a range of values, derived from the sample data, that is likely to contain the population parameter with a certain level of confidence.
What are ‘discrete probability distributions’ and their use?
Discrete probability distributions deal with outcomes that take specific values, such as integers, used in scenarios where outcomes are countable.
Define ‘empirical probability distribution’.
An empirical probability distribution is based on observed data, representing the relative frequency of outcomes in a sample.
Explain ‘mean’ as a measure of central tendency.
The mean is the average of a set of numbers, calculated by adding all the figures together and dividing by the count of numbers.
What is ‘mode’ in statistical terms?
The mode is the value that appears most frequently in a data set, representing the most typical value.
How does ‘curve fitting’ relate to regression analysis?
Curve fitting involves adjusting mathematical curves to fit a set of data points, with regression analysis being one method to achieve this.
What does ‘correlation coefficient’ indicate in a statistical model?
The correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables.
Discuss the importance of ‘least squares’ in data fitting.
Least squares is a method used in regression analysis to determine the best fit line
What defines an ‘analytic geometry’ problem?
Analytic geometry involves using algebraic methods to solve geometry problems, focusing on points, lines, and curves.
How is ‘differential calculus’ used to optimize functions?
Differential calculus is used to find the maximum and minimum values of functions by finding their derivatives and solving for zeros.
What is the fundamental theorem of calculus?
The fundamental theorem of calculus links the concept of differentiation and integration, showing that these two operations are inverses of each other.
How are functions of multiple variables integrated in multivariable calculus?
Functions of multiple variables are integrated using multiple integrals, which calculate the volume under the surface described by the function.
Describe a method to solve nonhomogeneous ordinary differential equations.
One method to solve nonhomogeneous ODEs is the method of undetermined coefficients, which finds a particular solution by assuming a form for the solution.
Explain how Laplace transforms are used in control systems.
Laplace transforms are used in control systems to convert differential equations of system dynamics into algebraic equations, simplifying the analysis and design of the systems.
What is the purpose of the dot product in physics?
The dot product is used in physics to calculate the magnitude of one vector in the direction of another, important in determining work done by a force.
Describe how matrix inversion is used in solving systems of equations.
Matrix inversion is used to solve systems of linear equations by finding the matrix inverse that, when multiplied with the coefficient matrix, yields the solution vector.
What role does error estimation play in numerical methods?
Error estimation provides an assessment of how far a numerical solution is likely to be from the true solution, guiding precision in computations.
What is a Fourier series and its application in engineering?
A Fourier series decomposes periodic functions into a sum of sines and cosines, used in engineering for signal processing and solving differential equations.
Define ‘binomial theorem’ in algebra.
The binomial theorem provides a formula for expanding powers of binomials, stating that (𝑎+𝑏)𝑛(a+b)n equals the sum of terms (𝑛𝑘)𝑎𝑛−𝑘𝑏𝑘(kn)an−kbk.
What are the properties of the normal distribution in statistics?
The normal distribution is symmetric about its mean, with its shape defined by the mean and the standard deviation, and is important in many statistical methods.
How is the ‘median’ different from the ‘mean’?
The median is the middle value in a data set when the values are arranged in order, whereas the mean is the average of all values.
Explain the role of ‘variance’ in probability and statistics.
Variance measures the spread of a set of numbers, calculating the average squared deviations from the mean, and is a foundational concept in statistical dispersion.
What is ‘multivariate regression analysis’?
Multivariate regression analysis involves predicting a dependent variable using multiple independent variables to explore complex relationships.
Describe ‘discrete’ versus ‘continuous’ probability distributions.
Discrete distributions describe outcomes with specific, countable values, while continuous distributions describe outcomes on a continuous scale.
What is ‘sample space’ in probability theory?
The sample space in probability theory is the set of all possible outcomes of a random experiment.
How are probability distributions used in risk analysis?
Probability distributions model the likelihood of different outcomes in risk analysis, helping to understand uncertainties and make informed decisions.
Explain ‘conditional probability’ and its importance.
Conditional probability measures the probability of an event occurring given that another event has already occurred, crucial for understanding dependent events.
What is the use of ‘expected value’ in insurance and finance?
In insurance and finance, the expected value is used to calculate the average outcome over the long term, guiding pricing and risk management strategies.
Define ‘least squares regression’ in simple terms.
Least squares regression is a statistical method that finds the best-fitting line through points by minimizing the sum of the squares of the vertical deviations from the points to the line.
How does ‘logistic regression’ differ from ‘linear regression’?
Logistic regression is used for binary outcomes and models probabilities of occurrence, whereas linear regression predicts continuous outcomes.
What is the ‘Poisson distribution’ and its typical applications?
The Poisson distribution models the probability of a given number of events happening in a fixed interval of time or space, used commonly in queuing theory.
Describe how ‘goodness of fit’ is used in model validation.
Goodness of fit tests assess how well the observed data match
A. Analytic geometry
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B. Calculus (e.g., differential, integral, single-variable, multivariable)
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C. Ordinary differential equations (e.g., homogeneous, nonhomogeneous,Laplace transforms)
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D. Linear algebra (e.g., matrix operations, vector analysis)
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E. Numerical methods (e.g., approximations, precision limits, error propagation, Taylor’s series, Newton’s method)
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F. Algorithm and logic development (e.g., flowcharts, pseudocode)
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A. Probability distributions (e.g., normal, binomial, empirical, discrete,continuous)
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B. Measures of central tendencies and dispersions (e.g., mean, mode,standard deviation, confidence intervals)
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C. Expected value (weighted average) in decision making
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D. Regression (linear, multiple), curve fitting, and goodness of fit(e.g., correlation coefficient, least squares)
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