In this lecture, we start our discussion of continuous random variables. We will focus on the case of a single continuous random variable, and we’ll describe its distribution using a so-called probability density function, an object that will replace the PMFs from the discrete case. We will then proceed to define the expectation and the …
Probability – L08.1 Lecture Overview – Continuous RV and PDFs Read More »
By now, we have defined the notion of independence of events and also the notion of independence of random variables. The two definitions look fairly similar, but the details are not exactly the same, because the two definitions refer to different situations. For two events, we know what it means for them to be independent. …
Probability – S07.3 Independence of Random Variables Versus Independence of Events Read More »
In this segment, we will derive the formula for the variance of the geometric PMF. The argument will be very much similar to the argument that we used to drive the expected value of the geometric PMF. And it relies on the memorylessness properties of geometric random variables. So let X be a geometric random …
Probability – S07.2 The Variance of the Geometric Read More »
In this segment, we develop the inclusion-exclusion formula, which is a beautiful generalization of a formula that we have seen before. Let us look at this formula and remind ourselves what it says. If we have two sets, A1 and A2, and we’re interested in the probability of their union, how can we find it? …
Probability – S07.1 The Inclusion-Exclusion Formula Read More »
We will now study a problem which is quite difficult to approach in a direct brute force manner but becomes tractable once we break it down into simpler pieces using several of the tricks that we have learned so far. And this problem will also be a good opportunity for reviewing some of the tricks …
Let us now revisit the variance and see what happens in the case of independence. Variances have some general properties that we have already seen. However, since we often add random variables, we would like to be able to say something about the variance of the sum of two random variables. Unfortunately, the situation is …
Probability – L07.7 Independence, Variances & the Binomial Variance Read More »
When we have independence, does anything interesting happen to expectations? We know that, in general, the expected value of a function of random variables is not the same as applying the function to the expected values. And we also know that there are some exceptions where we do get equality. This is the case where …
Probability – L07.6 Independence & Expectations Read More »
Let us now consider a simple example. Let random variables X and Y be described by a joint PMF which is the one shown in this table. Question– are X and Y independent? We can try to answer this question by using the definition of independence. But it is actually more instructive to proceed in …
We now come to a very important concept, the concept of independence of random variables. We are already familiar with the notion of independence of two events. We have the mathematical definition, and the interpretation is that conditional probabilities are the same as unconditional ones. Intuitively, when you are told that B occurred, this does …
Probability – L07.4 Independence of Random Variables Read More »
We will now talk about conditional expectations of one random variable given another. As we will see, there will be nothing new here, except for older results but given in new notation. Any PMF has an associated expectation. And so conditional PMFs also have associated expectations, which we call conditional expectations. We have already seen …
Probability – L07.3 Conditional Expectation & the Total Expectation Theorem Read More »