Visualizing the average precipitation among the US states

Problem Consider the following dataset containing the average annual precipitation in the US states between 1971-2000. Make a choropleth visualization of this precipitation data (either using the SI or the US-British units). Do not forget to add a color-bar to...

Computing the first four moments of a sample

Problem Consider this dataset comprised of $1000$ observations (tuples). Compute the first four standardized moments of this sample (mean, standard deviation, skewness, kurtosis).

An experimental proof of Chebyshev's inequality

Problem The Chebyshev Inequality states that no more than $1/k^2$ of an attribute values of a given sample can be $k$ or more standard deviations away from the attribute mean. Provide an experimental proof of this theorem by generating a...

Computing the mean of a weighted data

Problem Consider this weighted dataset comprised of $500$ observations (tuples) each of which is described by $5$ attributes. Note that the last column of data is the weight of each tuple. Compute the weighted mean of this sample.

Monte Carlo approximation of the number Pi

Problem Compute the following 10-dimensional integral via Monte Carlo Rejection sampling method, \[I = \int_{x_1 = 0}^{x_1 = 1} dx_1 \cdots \int_{x_{10} = 0}^{x_{10} = 1} dx_{10} \bigg(\sum_{i=1}^{i=10} ~ x_i ~ \bigg) ~,\] Ensure the accuracy of your integration result...

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....

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...