Monte Carlo approximation of the number Pi using a full circle

Problem Suppose we did not know the value of $\pi$ and we wanted to estimate its value using Monte Carlo methods. One practical approach is to draw a square sides equal to $a = 2$, with its diagonal opposite corners...

Monte Carlo approximation of the area of heart

Problem A popular mathematical equation for 2D heart is the following, \[f(x,y) = (x^2 + y^2 - 1)^3 - x^2 y^3 = 0\] Any $(x,y)$ values that result in $f(x,y) < 0$ represent the coordinates of a point that falls...

Parsing data from the World Wide Web

Consider the following web-page address This is a data table in HTML language containing data from the NASA Swift satellite. Each row in this table represents information about a Gamma-Ray Burst (GRB) detection that Swift has made in the...

Data transfer: Converting formatted input to Comma-Separated-Values (CSV) output

Problem Consider this formatted data file: Write a simple script named formatted2csv that takes two input arguments representing the input and output file names. Then, the script writes the same input float data to the output file data.out in...

Command line input option-value pairs

Problem Python Suppose we want to write a program that takes in three input parameters: the initial height ($y_0$) initHeight, the initial velocity ($v_0$) initVelocity, the time after which we want to know how much a projectile has moved in...

Python modules and packaging

Problem Consider the following codes that compute the Fibonacci sequence using two different methods: and Put these two functions in a folder named fib such that they can be imported as a Python package to your Python environment....

Visualization: The world population

Problem The following plot shows the worldwide population by countries and states. What kind of visualization is this plot?

Visualization: The world population (refined)

Problem The following plot is a refined map of the worldwide population. What kind of visualization and map is this plot? Select all that apply. Robinson map, Interrupted Goode Homolosine map, Cartesian Longitude and Latitude map, Cartogram Heat map, Winkel...

Visualization: The world map resized by population

Problem The following plot uses (distorts) the country sizes to represent the worldwide population. What kind of visualization (map) is this plot? Cartogram Heat map, Robinson map, Winkel map, Choropleth, Cartogram,

Visualization: Worldwide Internet Usage

Problem The following plot shows the worldwide usage of Internet by countries. What kind of visualization is this plot? (By the way, which country is the largest Internet-user in the world?)

Logic NAND equivalence

We have learned that, \[\begin{eqnarray} \bar{A} &=& A \uparrow A ~, \nonumber \\ AB &=& (A \uparrow B) \uparrow (A \uparrow B) ~, \nonumber \\ A + B &=& (A \uparrow A) \uparrow (B \uparrow B) ~, \nonumber \end{eqnarray}\] Now,...

The fundamental desiderata of Probability Theory

Name the three fundamental desiderata of Probability Theory.

Probability Theory: correspondence with commonsense

Show via an example Venn diagram that if, holds, then, also holds.

The proof of Bayes' Rule via Venn diagram

Prove the Bayes’s rule via Venn diagrams, \[\pi(B|A) = \frac{ \pi(A|B) ~ \pi(B) } { \pi(A) } ~.\]

The fundamental logical operators

Show that the following identities hold, where the two arrow-up and arrow-down symbols represent the fundamental NAND and NOR operators,

Logic functions in terms of logic functions

Show that the following functions, can be written as, where the basis logic functions have the following truth table,

Logic functions with 2 input

Consider the following special functions that are TRUE only at specific points within the logical sample space: Show that the above truth table is equivalent to the following logical operations.

Logic functions with 1 input

Consider a set of logic functions \(\{ f_1(A), f_2(A), f_3(A), f_4(A)\}\) that take a proposition (\(A\)) as input which is either True or False. These functions map the truth value of the input proposition to either True or False according...

The two types of scientific reasoning

Name the two types of scientific reasoning and give an example argument for each one.