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Taking free online courses is gaining momentum. There are many options to choose from: Khan Academy, Education – YouTube, and Udacity. Some come with ivy league and other top universities backing such as MIT’s and Harvard’s edX (a Joint venture builds on MITx and Harvard distance learning), and Coursera which offers host courses from Princeton University, Stanford University, University of California, Berkeley, University of Michigan-Ann Arbor, and University of Pennsylvania.
The world is watching and people have welcomed this movement with open arms! For instance, Udacity’s (founded by three roboticists) first class, “Introduction to Artificial Intelligence,” over 160,000 students in more than 190 countries enrolled! They aim to continue this trend by teaching 160,000 plus students statistics with it’s latest course: “Intro to Statistics (ST101) Making Decisions Based on Data.” This class is being taught by leading computer scientist and Udacity co-founder Sebastian Thurn a PhD in computer science and statistics.
Khan Academy is not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere.
Coursera is a social entrepreneurship company that partners with the top universities in the world to offer courses online for anyone to take, for free.
Udacity is a private educational organization that believes university-level education can be both high quality and low cost (For some courses, you will have an opportunity to take a certified exam, which will have some cost.)
I recently experienced three consecutive days that are major anniversaries for me: June 12 is my baptismal anniversary, June 13 is my birthday, and June 14 is my wedding anniversary (also my daughter turned 15 months old on this day). These events put to the forefront of my mind a famous problem in probability, the birthday problem.
— The Uniform Probability Model for a Finite Sample Space —
There are many different probability models. We’ll start with the simplest of these models, the uniform probability model for a finite sample space.
Let be a finite sample space. The uniform probability model for is defined by
as defined satisfies the axioms for a probability measure. By adopting this model, we are predicting that in a sequence of experiments with sample space the observed relative frequency of events in the sequence of experiments will be equal to the relative frequency, , of events in the set . In particular, for each outcome , we have , or as one says, “All outcomes in are equally likely (i.e., equally probable)." Incidentally, for events consisting of a single outcome are sometimes called elementary events.
assuming all tuples are equally likely. Below are some values of :
Remarkably, in a room with just people, the probability is greater than that two or more people have the same birthday!
— Binomial Random Variables and Poisson Approximations —
In the case of a binomial experiment there are independent trials, with the results of each trial labeled “success" or “failure." On each trial the probability of success is and (hence) the probability of failure is . The sample space consists of the set of words in the alphabet and the random variable on is defined by (each word) the number of ‘s in that word. The probablity density function (pdf) for is given by
We say that such a random variable is a binomial random variable with parameters and , abbreviating this with the notation . We also say that X has a binomial distribution with parameters and , and that is a binomial density function with parameters and . Note, in general, if , then .
If , where is “large" and is “small," then for
where . This is called the Poisson approximation (Denis Poisson, 1781-1840) to the binomial distribution.
The Poisson approximation of this quantity is