We now study a model that involves the sum of independent random variables, but with a twist. It’s going to be the sum of a random number of independent random variables, as opposed to a fixed number. This is a model that shows up in a variety of applications, but it will also help us …
Probability – L13.10 Mean of the Sum of a Random Number of Random Variables Read More »
We will now go through another example to consolidate our intuition about the content of the law of iterated expectations and the law of the total variance. The example is as follows. We have a class, and that class consists of 30 students in total who are divided into sections– the first and the second …
We will now go through an example, which is essentially a drill, to consolidate our understanding of the conditional expectation and the conditional variance. Consider a random variable X, which is continuous and is described by a PDF of this form. Whenever we have a PDF that seems to consist of different pieces, it’s always …
We will now go through a derivation of the law of total variance. This particular derivation is not insightful. It will not really give you any intuition as to why the law of total variance is correct. On the other hand, it involves some interesting manipulations that will be useful to be able to follow, …
Probability – L13.7 Derivation of the Law of Total Variance Read More »
We have defined the conditional expectation of a random variable given another as an abstract object, which is itself a random variable. Let us now do something analogous with the notion of [the] conditional variance. Let us start with the definition of the variance, which is the following. We look at the deviation of the …
This is an example, or rather, a story, that’s supposed to give us some insight and some intuition about what the law of iterated expectations really means. Suppose that you work for a forecasting company. And suppose that you make forecasts by calculating expected values of the quantities that you want to forecast. And of …
Here is a simple application of the law of iterated expectations. We revisit the stick-breaking example, which we have seen sometime in the past. So in this example, we start with a stick that has a certain length and which we break at a point that’s chosen uniformly at random throughout the length of the …
We have previously defined the abstract conditional expectation of one random variable given another random variable. And we discussed that it is, by itself, a random variable. In particular, it has an expectation, or mean, of its own. What is this mean? This is what we want to find out. Let us recall our development. …
Probability – L13.3 The Law of Iterated Expectations Read More »
Introduction In this segment we revisit the concept of conditional expectation and view it as an abstract object of a special kind. To get going, let us start with something simple, the concept of a function. Let’s say a function h that maps real numbers to real numbers. As a concrete instance, consider the quadratic …
Probability – L13.2 Conditional Expectation as a Random Variable Read More »
By this point in this class, you must have realized that a lot of revolves around the concept of conditioning. Conditional expectations play a central role. For this reason, it is useful to revisit this concept and view it in a more abstract manner. The basic idea is that the value of a conditional expectation …