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MITOCW | watch?v=f9XFM8YLccg The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, so good afternoon. Today, we will review probability theory. So I will mostly focus on-- I'll give you some distributions. So probabilistic distributions, that will be of interest to us throughout the course. And I will talk about moment-generating function a little bit. Afterwards, I will talk about law of large numbers and central limit theorem. Who has heard of all of these topics before? OK. That's good. And I'll try to focus more on a little bit more of the advanced stuff. Then a big part of it will be review for you. So first of all, just to agree on terminology, let's review some definitions. So a random variable x-- we will talk about discrete and continuous random variables. Just to set up the notation, I will write discrete at x and continuous random variable as y for now. So they are given by its probability distribution-- discrete random variable is given by its probability mass function. f sum x I will denote. And continuous is given by probability distribution function. I will denote by x sub y. So pmf and pdf. Here, I just use a subscript because I wanted to distinguish f of x and x of y. But when it's clear which random variable we're talking about, I'll just say f. So what is this? A probability mass function is a function from the sample space to non-negative reals such that the sum over all points in the domain equals 1. The probability distribution is very similar. The function from the sample space non- negative reals, but now the integration over the domain. So it's pretty much safe to consider our sample space to be the real numbers for continuous random variables. Later in the course, you will see some examples where it's not the real numbers. 1
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But for now, just consider it as real Numbers. For example, probability mass function. If X takes 1 with probability 1/3 minus 1 of probability 1/3 and 0 with probability 1/3. Then our probability mass function is fx 1 equals fx minus 1 1/3, just like that. An example of a continuous random variable is if-- let's say, for example, if x of y is equal to 1 for all y and 0,1, then this is pdf of uniform random variable where the space is 0. So this random variable just picks one out of the three numbers with equal probability. This picks one out of this. All the real numbers are between 0 and 1 with equal probability. These are just some basic stuff. You should be familiar with this, but I wrote it down just so that we agree on the notation.
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  • Spring '17
  • Jim Angel

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