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 »
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 …