In the discrete case, we saw that we could recover the PMF of X and the PMF of Y from the joint PMF. Indeed, the joint PMF is supposed to contain a complete probabilistic description of the two random variables. It is their probability law, and any quantity of interest can be computed if we …
Probability – L09.8 From The Joint to the Marginal Read More »
In this segment, we start a discussion of multiple continuous random variables. Here are some objects that we’re already familiar with. But exactly as in the discrete case, if we are dealing with two random variables, it is not enough to know their individual PDFs. We also need to model the relation between the two …
We now look at an example similar to the previous one, in which we have again two scenarios, but in which we have both discrete and continuous random variables involved. You have $1 and the opportunity to play in the lottery. With probability 1/2, you do nothing and you’re left with the dollar that you …
We now continue with the development of continuous analogs of everything we know for the discrete case. We have already seen a few versions of the total probability theorem, one version for events and one version for PMFs. Let us now develop a continuous analog. Suppose, as always, that we have a partition of the …
Probability – L09.5 Total Probability & Expectation Theorems Read More »
We now revisit the exponential random variable that we introduced earlier and develop some intuition about what it represents. We do this by establishing a memorylessness property, similar to the one that we established earlier in the discrete case for the geometric PMF. Suppose that it is known that light bulbs have a lifetime until …
Probability – L09.4 Memorylessness of the Exponential PDF Read More »
Let us now look at an example. Consider a piecewise constant PDF of the form shown in this diagram. Suppose that we condition on the event that x lies between a plus b over 2, which is here, and b. So we’re conditioning on x lying in this particular red interval. What is the conditional …
In this segment, we pursue two themes. Every concept has a conditional counterpart. We know about PDFs, but if we live in a conditional universe, then we deal with conditional probabilities. And we need to use conditional PDFs. The second theme is that discrete formulas have continuous counterparts in which summations get replaced by integrals, …
Probability – L09.2 Conditioning A Continuous Random Variable on an Event Read More »
In this lecture, we continue our discussion of continuous random variables. We will start by bringing conditioning into the picture and discussing how the PDF of a continuous random variable changes when we are told that a certain event has occurred. We will take the occasion to develop counterparts of some of the tools that …
Probability – L09.1 Lecture Overview – Conditioning on an event; Multiple RVs Read More »
We have claimed that normal random variables are very important, and therefore we would like to be able to calculate probabilities associated with them. For example, given a normal random variable, what is the probability that it takes a value less than 5? Unfortunately, there are no closed form expressions that can help us with …
Probability – L08.9 Calculation of Normal Probabilities Read More »
We now introduce normal random variables, which are also often called Gaussian random variables. Normal random variables are perhaps the most important ones in probability theory. They play a key role in the theory of the subject, as we will see later in this class in the context of the central limit theorem. They’re also …