Just in order to get some more familiarity with joint PDFs, let us look at independent normals. Actually, this is an important example because noise is often modeled by normal random variables, and noise terms that show up at different parts of a system, or at different times, are often assumed to be independent. Suppose …
We will now go through an example that brings together all of the concepts that we have introduced. We have a stick of length l. And we break that stick at some random location, which corresponds to a random variable, X. And we assume that this random variable is uniform over the length of the …
Independence is one of the central concepts of probability theory, because it allows us to build large models from simpler ones. How should we define independence in the continuous case? Our guide comes from the discrete definition. By analogy with the discrete case, we will say that two jointly continuous random variables are independent if …
Conditional PDFs share most of the properties of conditional PMFs. All facts for the discrete case have continuous analogs. The intuition is more or less the same, although it is much easier to grasp in the discrete case. For example, we have seen this version of the total probability theorem. There is a continuous analog …
Probability – L10.4 Total Probability & Total Expectation Theorems Read More »
The definition of the conditional PDF is very simple. It is just this formula, which is analogous to the one for the discrete case. In all respects– mathematical and intuitive– it is very similar to the conditional PMF. Even so, developing a solid grasp of this concept does take some further thinking, so we will …
Probability – L10.3 Comments on Conditional PDFs Read More »
Introduction By now, we have introduced all sorts of PMFs for the discrete case. The joint PMF, the conditional PMF– given an event– and the conditional PMF of one random variable given another. And we’re moving along with the program of defining analogous concepts for the continuous case. We have already discussed the joint PDF …
In this lecture we complete our discussion of multiple continuous random variables. In the first half, we talk about the conditional distribution of one random variable, given the value of another. We will see that the mechanics are essentially the same as in the discrete case. Here, we will actually face some subtle issues, because …
In this segment, we will look at the famous example, which was posed by Comte de Buffon– a French naturalist– back in the 18th century. And it marks the beginning of a subject that is known as the subject of geometric probability. The problem is pretty simple. We have the infinite plane, and we draw …
Probability – S09.1 Buffon’s Needle & Monte Carlo Simulation Read More »
Besides PMFs and PDFs, we can also describe the distribution of a random variable, as we know, using a CDF. A CDF is always well-defined. And for the case of a continuous random variable, the CDF can be found by integrating the PDF. And conversely, we can recover the PDF from the CDF by differentiating. …
In this segment we will go very fast through a few definitions and facts that remain true in the continuous case. Everything is completely analogous to the discrete case. And there are absolutely no surprises here. So, for example, we have defined joint PMFs for the case of more than two discrete random variables. And …
Probability – L09.9 Continuous Analogs of Various Properties Read More »