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 …
In this segment, we make a connection between the correlation coefficient and some fairly realistic real world situations. The bottom line will be that the presence or absence of correlations can make a huge difference. Suppose that you run an investment company that invests in real estate, and you have 100 million of capital that …
The mathematics of the correlation coefficient are important. But it is perhaps more important to be able to interpret it correctly. A correlation coefficient of let’s say 0. 5, tells us that something interesting is going on as far as the relation of X and Y is concerned. But what exactly? It tells us that …
Probability – L12.10 Interpreting the Correlation Coefficient Read More »
In this segment, we justify some of the property is that the correlation coefficient that we claimed a little earlier. The most important properties of the correlation coefficient lies between minus 1 and plus 1. We will prove this property for the special case where we have random variables with zero means and unit variances. …
Probability – L12.9 Proof of Key Properties of the Correlation Coefficient Read More »
The covariance between two random variables tells us something about the strength of the dependence between them. But it is not so easy to interpret qualitatively. For example, if I tell you that the covariance of X and Y is equal to 5, this does not tell you very much about whether X and Y …
One situation where covariances show up is when we try to calculate the variance of a sum of random variables. So let us look at the variance of the sum of two random variables, X1 and X2. If the two random variables are independent, then we know that the variance of the sum is the …
Probability – L12.7 The Variance of the Sum of Random Variables Read More »
In this segment, we discuss a few algebraic properties of the covariance. There is nothing deep here, only some observations that can be useful if we want to carry out covariance calculations. We start by looking at this quantity, the covariance of a random variable with itself. So it’s a special case of this definition …