An important reason why conditional probabilities are very useful is that they allow us to divide and conquer. They allow us to split complicated probability modes into simpler submodels that we can then analyze one at a time. Let me remind you of the Total Probability Theorem that has his particular flavor. Total Probability Theorem …
We now move to a new topic– conditioning. Every probabilistic concept or probabilistic fact has a conditional counterpart. As we have seen before, we can start with a probabilistic model and some initial probabilities. But then if we are told that the certain event has occurred, we can revise our model and consider conditional probabilities …
Probability – L06.4 Conditional PMFs & Expectations Given an Event Read More »
In this segment, we will go through the calculation of the variances of some familiar random variables, starting with the simplest one that we know, which is the Bernoulli random variable. So let X take values 0 or 1, and it takes a value of 1 with probability p. We have already calculated the expected …
Probability – L06.3 The Variance of the Bernoulli & The Uniform Read More »
We have introduced the concept of expected value or mean, which tells us the average value of a random variable. We will now introduce another quantity, the variance, which quantifies the spread of the distribution of a random variable. So consider a random variable with a given PMF, for example like the PMF shown in …
In the previous lecture we introduced random variables, probability mass functions and expectations. In this lecture we continue with the development of various concepts associated with random variables. There will be three main parts. In the first part we define the variance of a random variable, and calculate it for some of our familiar random …
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 …
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, …