Statistics Weekly HW

HW_1

Research
1. Definition of statistics
2. What is a dataset? From the observation units to the dataset attributes and values
3. Find intresting application of statistics in cybersecurity
Application
1. Create a simple application in C# and VB.net
Research on App
1. Main differences between C# and VB.net
Theory Link

Research
1. Definition of distribution
Application
1. Make a simple demo program which uses the objects RANDOM and TIMER in C# and VB.NET
2. Make a simple CSV parser that can ead a CSV file
3. Compute a univariate distribution of a variable after reading a variable from a CSV file
Research on App
1. Summarize the main aspects of the CSV protocol RFC 4180 (definition and rules)
Theory Link

Research
1. Illustrate the concept of conditional, joint, marginal (relative) frequency using a simple bivariate distribution
2. Illustrate the concept of statistical independence and the resulting mathematical relationships between the above frequencies
Application
1. Create a distribution from the data obtained by the sniffer Wireshark by reading the CSV file or realtime data generated by the program [optional: create a bivariate distribution]
Research on App
1. A survey on ONLINE algorithms (mean, variance, median, etc…)
2. Illustrate in particular, Knuth recursion for the computation of the arithmetic mean or average, discussion why it is preferable to the “naive” algo
Theory Link

Research
1. Illustrate the parallels, between the properties of the relative frequency and the axioms for probability
2. Discuss some concrete examples of Probability measure Space
3. Illustrate how Measure Theory provides the mathematical foundation for Probability Theory
Application
1. Generate a m sequences of n trials distributed according a Binomial with success probability p and represent the following quantities: absolute frequency of success, relative frequency of success, “normalized” relative frequency of success.
2. In the same chart, add also a vertical histogram representing the distribution of the above 3 types of frequencies on the last trial.
Research on App
1. Give a simple introduction to graphics in the .NET environment. How to create a bitmap and a chart on it.
2. Explain in simple terms how to get device coordinates from world coordinates
Theory Link

Research
1. Explain all possible derivation of the arithmetic mean and in general of the other common types of averages
2. Illustrate the difference between “mathematical convergence” and “convergence” in probability
3. Illustrate the differences between Descriptive Statistics and Inferential Statistics and the role of probability and probability distributions
Application
1. Given an arbitrary rectangle and a distribution, create the corresponding histogram or column chart, with both vertical and horizontal orientation
2. Given a rectangle, enable its movement/resize/zoom using a mouse (this will be the resizable where we will draw alll future charts)
Theory Link

Research
1. Try to explain in your own words, the concept of population and sampling distribution
2. Show the expected value and variance of the sampling mean and take a look at the same sampling variance
Application
1. Take any population and any variable X and draw m (large) samples of size n (as you like) and represent the following 2 distributions: Distribution of the Mean and Distribution of the Variance. Finally, for each distribution compute mean and variance and compare with the corrisponding values in the population.
Theory Link

Research
1. Try to understand the general idea of the Lebesgue-Stieltjes integral and why it is useful concept and notation in Theory of Probability
2. Explain in your own words the “Law of large numbers” and sketch a simple proof (Markov inequality, Čebyšëv inequality, … )
Application
1. Consider a general scheme ad the simulation of previous homework and simulate the distribution of p = SUM(xi) , where xi are Bernoulli(lambda/n), with success probability lambda /n, where lambda is a user constant (“arrival rate”). Also plot the distribution of the “interarrival times”. Finally, search on the web to understand what you have actually simulated (eg: https://www.probabilitycourse.com/chapter11/11_1_2_basic_concepts_of_the_poisson_process.php )
Theory Link

Research
1. Search on the web about possible derivation of the Normal Distribution
2. Search on the web about method to generate normal variate (eg Marsaglia method, etc.)
Application
1. Consider X (radius), Y( angle) uniform r.vs. and use them to plot points on a chart. Determine the empirical distribution of the Cartesian coordinates
2. Search for the methods to generate a NORMAL r.v. X , and simulate (Application) the following distribution : X, X², X/Y², X²/Y² , X/Y . And find in the web what are these distributions
Researches about theory relevant to applications
1. Find in the web what are the distributions that you just simulated
Theory Link