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3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
Lecture: Monte Carlo Methods
Monte Carlo methods rely on repeated random sampling to compute results. They generate random samples from a population according to a probability distribution and use them to obtain numerical results. The founders of the Monte Carlo method were J. von Neumann and S. Ulam during the Manhattan Project in the 1940s. Monte Carlo methods can be used to solve multidimensional integrals and have better convergence than classical numerical integration methods for dimensions greater than 4. The variance of Monte Carlo estimates decreases as 1/N, where N is the number of samples, resulting in slow convergence. Variance reduction techniques can improve the convergence rate.
Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
Data Representation - Floating Point
This document discusses floating point numbers and how they are represented in computers using binary rather than decimal. It notes that floating point numbers are approximations and certain calculations may result in numbers like 0.999999999 instead of the expected 1. This is due to the limitations of binary representation. It then explains how floating point numbers use a significand and exponent to represent a wide range of values with limited precision. Finally, it provides some examples of converting between decimal and IEEE 754 single-precision floating point format.
Simulation
Simulation involves imitating the performance of a system using a model to generate sample experiments without observing the real system. Monte Carlo simulation specifically involves an element of chance and is useful when direct experimentation is impossible or too costly. It generates random numbers that can represent values of random variables to simulate observations. Common techniques include using uniform random numbers between 0 and 1 to represent distributions and the Box-Muller method for generating normal random variables.
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This document introduces machine learning and supervised learning. It discusses learning a classifier from labeled examples to predict a target variable. The key points covered are:
- Supervised learning involves learning a function that maps inputs to outputs from example input-output pairs.
- The goal is to learn a hypothesis h that has low error on the training set and generalizes well to new examples.
- The version space is the set of all hypotheses consistent with the training data.
- Controlling the complexity of the hypothesis class H via measures like VC dimension can improve generalization.
- For classification, multiple target classes are handled by learning one hypothesis per class. Regression learns a real-valued target function.
- There is a trade
Monte carlo simulation
This document discusses using Monte Carlo simulation to price European call options. It begins by noting the uncertainty in underlying asset prices makes option pricing difficult. It then outlines the Monte Carlo simulation procedure of generating random price paths, calculating payoffs, and averaging. Key assumptions are presented for the stock price model and payoff calculation. Simulation results are shown for different numbers of simulations and compared to the Black-Scholes model, with error decreasing as the number of simulations increases.
Monte Carlo Methods
This document discusses Monte Carlo methods for numerical integration and simulation. It introduces the challenge of sampling from probability distributions and several Monte Carlo techniques to address this, including importance sampling, rejection sampling, and Metropolis-Hastings. It provides pseudocode for rejection sampling and discusses its application to estimating pi. Finally, it outlines using Metropolis-Hastings to simulate the Ising model of magnetization.
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3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
Lecture: Monte Carlo Methods
Monte Carlo methods rely on repeated random sampling to compute results. They generate random samples from a population according to a probability distribution and use them to obtain numerical results. The founders of the Monte Carlo method were J. von Neumann and S. Ulam during the Manhattan Project in the 1940s. Monte Carlo methods can be used to solve multidimensional integrals and have better convergence than classical numerical integration methods for dimensions greater than 4. The variance of Monte Carlo estimates decreases as 1/N, where N is the number of samples, resulting in slow convergence. Variance reduction techniques can improve the convergence rate.
Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
Data Representation - Floating Point
This document discusses floating point numbers and how they are represented in computers using binary rather than decimal. It notes that floating point numbers are approximations and certain calculations may result in numbers like 0.999999999 instead of the expected 1. This is due to the limitations of binary representation. It then explains how floating point numbers use a significand and exponent to represent a wide range of values with limited precision. Finally, it provides some examples of converting between decimal and IEEE 754 single-precision floating point format.
Simulation
Simulation involves imitating the performance of a system using a model to generate sample experiments without observing the real system. Monte Carlo simulation specifically involves an element of chance and is useful when direct experimentation is impossible or too costly. It generates random numbers that can represent values of random variables to simulate observations. Common techniques include using uniform random numbers between 0 and 1 to represent distributions and the Box-Muller method for generating normal random variables.
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This document introduces machine learning and supervised learning. It discusses learning a classifier from labeled examples to predict a target variable. The key points covered are:
- Supervised learning involves learning a function that maps inputs to outputs from example input-output pairs.
- The goal is to learn a hypothesis h that has low error on the training set and generalizes well to new examples.
- The version space is the set of all hypotheses consistent with the training data.
- Controlling the complexity of the hypothesis class H via measures like VC dimension can improve generalization.
- For classification, multiple target classes are handled by learning one hypothesis per class. Regression learns a real-valued target function.
- There is a trade
Monte carlo simulation
This document discusses using Monte Carlo simulation to price European call options. It begins by noting the uncertainty in underlying asset prices makes option pricing difficult. It then outlines the Monte Carlo simulation procedure of generating random price paths, calculating payoffs, and averaging. Key assumptions are presented for the stock price model and payoff calculation. Simulation results are shown for different numbers of simulations and compared to the Black-Scholes model, with error decreasing as the number of simulations increases.
Monte Carlo Methods
This document discusses Monte Carlo methods for numerical integration and simulation. It introduces the challenge of sampling from probability distributions and several Monte Carlo techniques to address this, including importance sampling, rejection sampling, and Metropolis-Hastings. It provides pseudocode for rejection sampling and discusses its application to estimating pi. Finally, it outlines using Metropolis-Hastings to simulate the Ising model of magnetization.
Monte carlo
Monte Carlo methods use random sampling to solve problems numerically. They work by setting up probabilistic models and running simulations using random numbers. This allows approximating solutions to problems in physics, finance, optimization, and other fields. Examples include estimating pi by simulating dart throws, and using a "drunken wino" random walk simulation to approximate the solution to a partial differential equation on a grid. The accuracy of Monte Carlo methods increases with more simulation iterations, requiring truly random numbers for best results.
Monte carlo simulation
Monte Carlo simulation is a statistical technique that uses random numbers and probability to simulate real-world processes. It was developed in the 1940s by scientists working on nuclear weapons research. Monte Carlo simulation provides approximate solutions to problems by running simulations many times. It allows for sensitivity analysis and scenario analysis. Some examples include estimating pi by randomly generating points within a circle, and approximating integrals by treating the area under a curve as a target for random darts. The technique provides probabilistic results and allows modeling of correlated inputs.
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The document discusses key concepts in probability and statistics including mean, variance, and standard deviation. It provides a 5-step process to calculate variance which involves finding the mean, subtracting it from each value, squaring the differences, multiplying by the probability, and summing the results. The standard deviation is defined as the square root of the variance.
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This document discusses finding extreme values and critical points of functions by examining their first and second derivatives. It provides examples of using the first derivative test to find local maximum and minimum values when the derivative is equal to zero or undefined. The second derivative test is introduced to determine concavity and relative maximum or minimum values based on whether the second derivative is positive, negative, or zero. Graphs, sign lines, and examples are provided to demonstrate these concepts.
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2) An example of using the method to find the root of the quadratic equation 2x^2 - 4x + 1 = 0. It shows calculating successive approximations that converge to the root.
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The monkey can move one space at a time in any cardinal direction on a planar grid. For any point on the grid, if the sum of the digits of the absolute value of the x coordinate plus the sum of the digits of the absolute value of the y coordinate is less than or equal to 19, the monkey can access that point. Starting from (0,0), the question asks how many points on the grid the monkey can access given this rule.
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The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
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What is the probability of winning the lottery 6/49, that is, picking the correct set of six numbers out of 49?
Get your answer here.......
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It is a global network of computers
that communicate with each other
using a variety of protocols and
overcoming various communication
barriers.
Bluetooth mobileip
Consortium: Ericsson, Intel, IBM, Nokia, Toshiba…
Scenarios:
connection of peripheral devices
loudspeaker, joystick, headset
support of ad-hoc networking
small devices, low-cost
bridging of networks
e.g., GSM via mobile phone - Bluetooth - laptop
Simple, cheap, replacement of IrDA, low range, lower data rates, low-power
Worldwide operation: 2.4 GHz
Resistance to jamming and selective frequency fading:
FHSS over 79 channels (of 1MHz each), 1600hops/s
Coexistence of multiple piconets: like CDMA
Workspace design
This document discusses the workspace design of robots. It defines key robot terminology like degree of freedom (DOF) and provides examples of different joint types and their DOFs, such as pin joints with 1 DOF and ball and socket joints with 3 DOFs. It then explains how to understand a robot's workspace using the example of a cylindrical robot with 3 DOFs (linear motion along z and y axes and rotation along z-axis). The workspace is formed by tracing the path of the end effector as it moves through its range of motion. Understanding the robot's workspace is important to avoid collisions when programming motions in industrial environments with multiple robots and components.
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Monte Carlo methods use random sampling to solve problems numerically. They work by setting up probabilistic models and running simulations using random numbers. This allows approximating solutions to problems in physics, finance, optimization, and other fields. Examples include estimating pi by simulating dart throws, and using a "drunken wino" random walk simulation to approximate the solution to a partial differential equation on a grid. The accuracy of Monte Carlo methods increases with more simulation iterations, requiring truly random numbers for best results.
Monte carlo simulation
Monte Carlo simulation is a statistical technique that uses random numbers and probability to simulate real-world processes. It was developed in the 1940s by scientists working on nuclear weapons research. Monte Carlo simulation provides approximate solutions to problems by running simulations many times. It allows for sensitivity analysis and scenario analysis. Some examples include estimating pi by randomly generating points within a circle, and approximating integrals by treating the area under a curve as a target for random darts. The technique provides probabilistic results and allows modeling of correlated inputs.
Statistics Powerpoint Standard Dev.
The document discusses key concepts in probability and statistics including mean, variance, and standard deviation. It provides a 5-step process to calculate variance which involves finding the mean, subtracting it from each value, squaring the differences, multiplying by the probability, and summing the results. The standard deviation is defined as the square root of the variance.
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This document discusses finding extreme values and critical points of functions by examining their first and second derivatives. It provides examples of using the first derivative test to find local maximum and minimum values when the derivative is equal to zero or undefined. The second derivative test is introduced to determine concavity and relative maximum or minimum values based on whether the second derivative is positive, negative, or zero. Graphs, sign lines, and examples are provided to demonstrate these concepts.
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This document contains 10 multiple choice questions about C programming fundamentals asked in an exam at Dezyne E'Cole College in Ajmer. The questions cover topics like data types, operators, pointers, structures and memory allocation. The answers provided explain the output or correct the errors in code snippets related to each question.
Monte carlo integration, importance sampling, basic idea of markov chain mont...
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This document summarizes Monte Carlo integration techniques, including importance sampling and Markov chain Monte Carlo. It defines Monte Carlo integration as a numerical integration method using random numbers. The techniques were originally developed around 1944 to study the atomic bomb. Common uses are for evaluating integrals. The document outlines the basic steps of Monte Carlo methods and defines importance sampling as estimating properties of one distribution using samples from another similar distribution to reduce variance. It also defines Markov chain Monte Carlo as using a Markov chain to simulate samples from a target posterior probability distribution.Algorithm of some numerical /computational methods
Euler's method, Newton Raphson method, Guass-seidal Method, Trapezoidal method & Bisection Method
Algorithm and Flowchart.
Or ppt,new
This document discusses methods of simulation and Monte Carlo simulation. It is authored by a group including Roy Thomas, Sam Scaria, Sonu Sebastian, and others. The document defines simulation as using a model of a real system to conduct experiments on a computer in order to describe, explain, and predict the behavior of the real system. Monte Carlo simulation is described as using probability and sampling to solve complicated equations. Key steps of Monte Carlo simulation include drawing a flow diagram, determining variable distributions, selecting random numbers, and applying mathematical functions to obtain solutions. Examples of applications include queuing problems, inventory problems, and risk analysis.
5.3 Basic functions. A handout.
Selected items from the set theory and methodology and philosophy of mathematics and computer programming.
The monte carlo method
The Monte Carlo method uses random numbers and probability statistics to solve complex problems through approximation. It was developed in the 1940s by physicists working on nuclear weapons who needed to model radiation shielding. The method involves defining a domain of possible inputs, randomly generating inputs from that domain, performing deterministic computations using the inputs, and aggregating the results. Monte Carlo methods are useful when problems cannot be solved analytically due to uncertainty. They are commonly used to value options and other financial derivatives.
Iterative1
The document describes the iterative method for finding roots of equations. It discusses:
1) How to set up an iterative process to find a root by taking an initial approximation x0 and repeatedly applying the function pi(x) to get closer approximations x1, x2, etc. until the change between successive values is very small.
2) An example of using the method to find the root of the quadratic equation 2x^2 - 4x + 1 = 0. It shows calculating successive approximations that converge to the root.
3) How the iterative method can be implemented in C++ using a function that takes the initial value and returns the updated value at each iteration until the change is negligible.
Monte Carlo Simulation - Paul wilmott
The document discusses the history and development of Monte Carlo simulation methods in financial engineering. Some key points:
1) Monte Carlo simulation techniques originated from games of chance and probabilistic concepts in the 17th century. They were later applied to calculating integrals and solving differential equations.
2) In the 1940s/50s, the techniques were developed and applied at Los Alamos National Laboratory, coining the term "Monte Carlo."
3) In the 1970s, Monte Carlo methods became widely used in finance, with Black-Scholes options pricing and models simulating random asset price movements. They allow calculating expected option payoffs and fair values.
Mws gen inp_ppt_direct
The document discusses direct interpolation methods, including linear, quadratic, and cubic interpolation. It provides examples of using each method to find the velocity of a rocket at different times. Linear interpolation found a velocity of 393.7 m/s at t=16s, while quadratic and cubic interpolation found velocities of 392.19 m/s and 392.06 m/s respectively. The cubic method had the lowest relative error compared to the other results. The document also calculates the distance traveled and acceleration of the rocket using the cubic interpolation formula.
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This document provides an introduction to machine learning and supervised learning. It discusses key concepts like training examples, hypotheses, error, model complexity, and overfitting. The goal of supervised learning is to learn a function that maps inputs to outputs from example input-output pairs. Choosing the right model complexity involves balancing training error, generalization error, and model interpretability.
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This document discusses numerically solving a differential equation problem to calculate the value function y' = x-y, y(0) = 1 from x = 0 to x = 5 over 50 steps. The analytical solution is provided as 2e−x +x-1 for comparison.
1 mathematical challenge_problem
The monkey can move one space at a time in any cardinal direction on a planar grid. For any point on the grid, if the sum of the digits of the absolute value of the x coordinate plus the sum of the digits of the absolute value of the y coordinate is less than or equal to 19, the monkey can access that point. Starting from (0,0), the question asks how many points on the grid the monkey can access given this rule.
Day 1a examples
The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
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“A robot is a reprogrammable, multifunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks.”
Probability
What is the probability of winning the lottery 6/49, that is, picking the correct set of six numbers out of 49?
Get your answer here.......
Java
It is a global network of computers
that communicate with each other
using a variety of protocols and
overcoming various communication
barriers.
Bluetooth mobileip
Consortium: Ericsson, Intel, IBM, Nokia, Toshiba…
Scenarios:
connection of peripheral devices
loudspeaker, joystick, headset
support of ad-hoc networking
small devices, low-cost
bridging of networks
e.g., GSM via mobile phone - Bluetooth - laptop
Simple, cheap, replacement of IrDA, low range, lower data rates, low-power
Worldwide operation: 2.4 GHz
Resistance to jamming and selective frequency fading:
FHSS over 79 channels (of 1MHz each), 1600hops/s
Coexistence of multiple piconets: like CDMA
Workspace design
This document discusses the workspace design of robots. It defines key robot terminology like degree of freedom (DOF) and provides examples of different joint types and their DOFs, such as pin joints with 1 DOF and ball and socket joints with 3 DOFs. It then explains how to understand a robot's workspace using the example of a cylindrical robot with 3 DOFs (linear motion along z and y axes and rotation along z-axis). The workspace is formed by tracing the path of the end effector as it moves through its range of motion. Understanding the robot's workspace is important to avoid collisions when programming motions in industrial environments with multiple robots and components.
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Sfs4e ppt 05
This document contains a chapter about probability from a Pearson textbook. It includes objectives, definitions, examples, and explanations about key concepts in probability such as:
- The addition rule for disjoint (mutually exclusive) events, which states that the probability of event A or B occurring is equal to the sum of the individual probabilities if A and B cannot both occur.
- Subjective probabilities, which are based on personal judgment rather than experimental data.
- Using simulations and empirical methods to estimate probabilities based on observed frequencies from repeated experiments.
Pre-Cal 40S May 27, 2009
The document provides an introduction to basic probability concepts including:
- Sample space is the set of all possible outcomes of an experiment.
- An event is a subset of outcomes within the sample space.
- Probability is expressed as a ratio of favorable outcomes to total outcomes.
- Probability ranges from 0 to 1, with 0 being impossible and 1 being certain.
- Complementary events have probabilities that sum to 1.
7-Experiment, Outcome and Sample Space.pptx
1) 1.56 and 1.0 cannot be probabilities because probabilities must be between 0 and 1.
2) 0.46, 0.09, 0.96, 0.25, 0.02 can be probabilities because they are between 0 and 1.
3) a) The probability of obtaining a number less than 4 is 3/6 = 1/2
b) The probability of obtaining a number between 3 and 6 is 4/6 = 2/3
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Probability
- 1. Now it’s time to look at… Discrete Probability CMSC 203 Discrete Structures 1
- 2. Discrete Probability Everything you have learned about counting constitutes the basis for computing the probability of events to happen. In the following, we will use the notion experiment for a procedure that yields one of a given set of possible outcomes. This set of possible outcomes is called the sample space of the experiment. An event is a subset of the sample space. CMSC 203 Discrete Structures 2
- 3. Discrete Probability If all outcomes in the sample space are equally likely, the following definition of probability applies: The probability of an event E, which is a subset of a finite sample space S of equally likely outcomes, is given by p(E) = |E|/|S|. Probability values range from 0 (for an event that will never happen) to 1 (for an event that will always happen whenever the experiment is carried out). CMSC 203 Discrete Structures 3
- 4. Discrete Probability Example I: An urn contains four blue balls and five red balls. What is the probability that a ball chosen from the urn is blue? Solution: There are nine possible outcomes, and the event “blue ball is chosen” comprises four of these outcomes. Therefore, the probability of this event is 4/9 or approximately 44.44%. CMSC 203 Discrete Structures 4
- 5. Discrete Probability Example II: What is the probability of winning the lottery 6/49, that is, picking the correct set of six numbers out of 49? Solution: There are C(49, 6) possible outcomes. Only one of these outcomes will actually make us win the lottery. p(E) = 1/C(49, 6) = 1/13,983,816 CMSC 203 Discrete Structures 5
- 6. Complimentary Events Let E be an event in a sample space S. The probability of an event –E, the complimentary event of E, is given by p(-E) = 1 – p(E). This can easily be shown: p(-E) = (|S| - |E|)/|S| = 1 - |E|/|S| = 1 – p(E). This rule is useful if it is easier to determine the probability of the complimentary event than the probability of the event itself. CMSC 203 Discrete Structures 6
- 7. Complimentary Events Example I: A sequence of 10 bits is randomly generated. What is the probability that at least one of these bits is zero? Solution: There are 210 = 1024 possible outcomes of generating such a sequence. The event –E, “none of the bits is zero”, includes only one of these outcomes, namely the sequence 1111111111. Therefore, p(-E) = 1/1024. Now p(E) can easily be computed as p(E) = 1 – p(-E) = 1 – 1/1024 = 1023/1024. CMSC 203 Discrete Structures 7
- 8. Complimentary Events Example II: What is the probability that at least two out of 36 people have the same birthday? Solution: The sample space S encompasses all possibilities for the birthdays of the 36 people, so |S| = 36536. Let us consider the event –E (“no two people out of 36 have the same birthday”). –E includes P(365, 36) outcomes (365 possibilities for the first person’s birthday, 364 for the second, and so on). Then p(-E) = P(365, 36)/36536 = 0.168, so p(E) = 0.832 or 83.2% CMSC 203 Discrete Structures 8