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MIT 6.00SC | Recitation 07 | Distributions, Monte Carlo, and Regressions

Distribution

We have 3 types of Distribution which we have talked about:-

Uniform Distribution

Normal Distribution

Exponential Distribution

Uniform Distribution

A Uniform Distribution will look like this:-

The probability will be defined as 1/b-a

We can plot two type of graph for uniform distribution, and there respective plots

Continuous Graph

Discrete Graph.

Continuous Graph

Discrete Graph

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#Continuous Graph #Discrete Graph #Distribution #Estimation of PI #Exponential Distribution #Mean #Monte Carlo #Monty Hall Problem #Normal Distribution #Numeric Integration #Regressions #Standard Deviation #Uniform Distribution

MIT 6.00SC | Lecture 17 | Curve Fitting

Introduction

In the last lecture, when we were finding the value of Pi, we started with an assumption that if we drop large enough pins and if we repeat this experiment multiple times, if we get a very small standard deviation we will have a correct value of PI.

But this assumption is not correct. This was because we were believing that a statistically sound argument is equivalent to truth.

The…

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#Buffon Laplace Math #Curve Fitting #Error Modeling #Hook&039;s Law #Least Square Fit #Objective Function

MIT 6.00SC | Lecture 16 | Using Randomness to Solve Non-random Problems

Computational Models

We have seen some models of real world in past lecture, Gaussian or Normal Distribution is a great way to represent real world models with the help of Mean and Standard deviation.

A Normal Distribution can be fully characterized by its mean and standard deviation. This concept of characterization of a curve based on some parameters help in fully modeling real world system.

Mo…

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#Computational Models #Exponential Distribution #Integration #Monte Carlo Pi #Monty Hall #Pi Estimation #Uniform Distribution

MIT 6.00SC | Recitation 06 | Probability and Statistics

Probability of Binary Variable

A binary variable which can have only 2 possible outcome, like heads or tails, 0 or 1, on or off.

Lets say we have a variable A which can take values Heads or Tails, so the p(A=H) i.e., probability of A taking the value H.

An important thing to note about probability is that it lies between 0 and 1, i.e. 0 <= p(A=H) <= 1. This range of 0 to 1is continuous and…

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#Concrete Example of Probability #Monte Carlo Simulation #Probability of 2 Independent Events #Probability of Binary Variable

MIT 6.00SC | Lecture 15 | Statistical Thinking

Introduction

At the end of lecture 14, we were flipping coin, and trying to find number of sample which is enough to safely say that the probability of getting a head or tails after n trails is 0.5.

So to solve this, we can flip a coin just 2 times and get head and tails, now with this we will have the probability of 0.5 which is the correct answer, but the sample size is not good enough,…

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#Co-efficient of variation #Confidence Intervals.#Normal Distribution #Standard Deviation #Standard Error #Statistical Thinking #Variance

MIT 6.00SC | Lecture 14 | Sampling and Monte Carlo Simulation

Data Visualization

Consider the below example code:-

import pylab principal = 10000 #initial investment interestRate = 0.05 years = 20 values = [] for i in range(years + 1): values.append(principal) principal += principal*interestRate pylab.plot(values) pylab.show()

The above code will show you the power of compounding in investment, but the graph plotted with the above code is useless because…

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#Bernoulli’s Law of Large Numbers #Data Visualization #Gambler’s fallacy #Inferential Statistics #Monte Carlo Simulation #Null hypothesis #Randomness and Probability

MIT 6.00SC | Lecture 13 | Some Basic Probability and Plotting Data

Introduction

In the last lecture, we had a error in our code, because the Random walk code, did not output correct value of small samples as we had manually checked.

The problem was in the simWalks() method, which used wrong arguments, we had used this:-

distances.append(walk(f, homer, numTrials))

But we should have used:-

distances.append(walk(f, homer, numSteps))

So the complete corrected…

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#Causal Non Determinism #Copenhagen Doctrines #Data Visualization #Predictive Non Determinism #PyLab #Random Walk #Stochastic Process

MIT 6.00SC | Lecture 12 | Introduction to Simulation and Random Walks

Generator Yield

We will start with the example in the previous lecture, see the code here;-

import datetime class Person(object): def __init__(self, name): #Create a person with name self.name = name try: firstBlank = name.rindex(' ') # print "__init__: firstBlank: ", firstBlank self.lastName = name[firstBlank+1:] except : self.lastName = name self.birthDay = None def getLastName(self): #returns…

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#Brownian Motion #Computational Model #Generator Yield #Random Walk

MIT 6.00SC | Recitation 05 | Quiz 1 Answers and Object-Oriented Programming

The first 25 min of the Recitation is about Quiz 1 which we will not listen to now.

Object Oriented Programming

Classes allow to define a custom type. It allows to group data (attributes) and methods together, which is called encapsulation. We were already using inbuilt classes like int, float, dict etc.

Classes also allows for Inheritance and Polymorphism. Polymorphism allows to expect the same…

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#Encapsulation #Object Oriented Programming #Polymorphism #python

MIT 6.00SC |Lecture 11 | OOP and Inheritance

Introduction

Object oriented programming started with the development of these 2 programming language.

CLU

Smalltalk

These languages were not able to propel the OOP concepts till the advent of Java , followed by C++ and Python.

Abstract Data Type

The core of all Object Oriented programming is Abstract Data Type. The fundamental use of Abstract data type is that we can add user defined data type…

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#Abstract Data Type #Class Variable #Data Hiding #Inheritance #instance variables #Interface

MIT 6.00SC | Lecture 10 | Hashing and Classes

Hashing

Hashing is the concepts used in Dictionary are implemented in Python. This provides us with a very efficient search mechanism, but at the cost of space(Memory).

For understanding the concepts of hashing we will consider that we will use hashing on a set of integers.

Let consider a integer i, which we pass it to a hash() function, it generates another integer in some range.

hash(i) ---->…

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#bucket #built-in classes.#class #collision #exceptions #Hashing #linear rehash #modules

MIT 6.00SC | Lecture 09 | Memory and Search Methods

Memory

When we have an list of int’s as shown below

L = [35, 4, 5, 29, 17, 58, 0]

The access to this list is always constant time, for a simple reason being the size of int is constant for a platform and a programming language.

To see how this constant time access works, lets consider ints occupy 4 units of memory, this units can be 8 bits, 16 bits. So how to access the ith element of the list…

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#Amortized Complexity #List in Python #Merge Sort #Selection Sort

MIT 6.00SC | Optional Recitation | Algorithm Complexity and Class Review

To get a clear picture of this Recitation, kindly read the Recitation handout (PDF).

The first question to ask ourself is What is Big-O notation?

Big-O notation gives us a upper bound on how long somethings will execute. The important thing to remember is that this is not time bound. So Big-O does not take how long a program will execute, it tells how many steps it will take to execute a program.

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#Big O notation #Factorial #Recursion #Some Common Big-O Notations

MIT 6.00SC | Lecture 08 | Efficiency and Order of Growth

Efficiency

We should start this lecture with a very intrinsic question, Why is efficiency important?

With the advancement in the computing power we can solve most of computational problem using brute force, but as we all know some of the computational problem are not large, they are enormous, and it takes approximately around time at the order of days to solve them. So efficiency is the…

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#Average case #Best case #Big O notation #efficiency #order of growth #Random Access Model #Worst Case

MIT 6.00SC | Recitation 04 | Recursion, Pseudo code and Debugging

Recursion

We will start from the last recitation’s topic, Recursion. So as a review, can we define what is Recursion?

Recursion is a divide and conquer techniques, in which we first identify the base case, where we can find the answer immediately, and then we break the big problem into problem of smaller size of the same type.

The biggest question with recursion is when to use recursion and when…

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#debugging #Default Parameters #Pseudo code #Recursion

MIT 6.00SC | Lecture 07 | Debugging

Introduction

In this lecture we will main cover these topics:-

Floating Point Numbers

Debugging

Floating Point Numbers

To start understanding of Floating point numbers, we should all understand how a number is represented in computers, i.e. Binary Numbers.

The understanding of Binary numbers is crucial for Floating point numbers.

Binary Numbers

We are all familiar with the decimal number system,…

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#approximations #Binary #debugging #float #floating point

MIT 6.00SC | Lecture 06 | Recursion

Dictionaries

As we have seen in the previous lecture notes about Dictionaries, It is a <Key, Value> pair. The interesting thing about this is, what things can become keys and which things can become values.

So some examples of such association for Keys.

Keys with numbers

Keys with Strings

Keys with other objects including Dictionaries.

Along with the above property, you can also have great…

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#Dictionaries #Divide and Conquer #Exponentiation #Modular Abstraction #Palindrome #Recursion #Tower of Hanoi

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