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 »
We have already introduced the concept of the conditional PMF of a random variable, X, given an event A. We will now consider the case where we condition on the value of another random variable Y. That is, we let A be the event that some other random variable, Y, takes on a specific value, …
In this last lecture of this unit, we continue with some of our earlier themes, and then introduce one new notion, the notion of independence of random variables. We will start by elaborating a bit more on the subject of conditional probability mass functions. We have already discussed the case where we condition a random …
Let us now revisit the subject of expectations and develop an important linearity property for the case where we’re dealing with multiple random variables. We already have a linearity property. If we have a linear function of a single random variable, then expectations behave in a linear fashion. But now, if we have multiple random …
Probability – L06.8 Linearity of Expectations & The Mean of the Binomial Read More »
By this point, we have discussed pretty much everything that is to be said about individual discrete random variables. Now let us move to the case where we’re dealing with multiple discrete random variables simultaneously, and talk about their distribution. As we will see, their distribution is characterized by a so-called joint PMF. So suppose …
Probability – L06.7 Joint PMFs and the Expected Value Rule Read More »