We end this lecture sequence with the most important property of expectations, namely linearity. The idea is pretty simple. Suppose that our random variable, X, is the salary of a random person out of some population. So that we can think of the expected value of X as the average salary within that population. And …
Probability – L05.11 Linearity of Expectations Read More »
In this segment, we discuss the expected value rule for calculating the expected value of a function of a random variable. It corresponds to a nice formula that we will see shortly, but it also involves a much more general idea that we will encounter many times in this course, in somewhat different forms. Here’s …
We now note some elementary properties of expectations. These will be some properties that are extremely natural and intuitive, but even so, they are worth recording. The first property is the following. If you have a random variable which is non-negative, then its expected value is also non-negative. What does it mean that the random …
Probability – L05.9 Elementary Properties of Expectation Read More »
Our discussion of random variable so far has involved nothing but standard probability calculations. Other than using the PMF notation, we have done nothing new. It is now time to introduce a truly new concept that plays a central role in probability theory. This is the concept of the expected value or expectation or mean …
The last discrete random variable that we will discuss is the so-called geometric random variable. It shows up in the context of the following experiment. We have a coin and we toss it infinitely many times and independently. And at each coin toss we have a fixed probability of heads, which is some given number, …
Probability – L05.7 Geometric Random Variables Read More »
The next random variable that we will discuss is the binomial random variable. It is one that is already familiar to us in most respects. It is associated with the experiment of taking a coin and tossing it n times independently. And at each toss, there is a probability, p, of obtaining heads. So the …
Discrete Uniform. In this segment and the next two, we will introduce a few useful random variables that show up in many applications– discrete uniform random variables, binomial random variables, and geometric random variables So let’s start with a discrete uniform. A discrete uniform random variable is one that has a PMF of this form. …
We now want to introduce some examples of random variables, and we will start with the simplest conceivable random variable– a random variable that takes the values of 0 or 1, with certain given probabilities. Such a random variable is called a Bernoulli random variable. And the distribution of this random variable is determined by …
Probability – L05.4 Bernoulli & Indicator Random Variables Read More »
A random variable can take different numerical values depending on the outcome of the experiment. Some outcomes are more likely than others, and similarly some of the possible numerical values of a random variable will be more likely than others. We restrict ourselves to discrete random variables, and we will describe these relative likelihoods in …
Probability – L05.3 Probability Mass Functions Read More »
We will now define the notion of a random variable. Very loosely speaking, a random variable is a numerical quantity that takes random values. But what does this mean? We want to be a little more precise and I’m going to introduce the idea through an example. Suppose that our sample space is a set …
Probability – L05.2 Definition of Random Variables Read More »