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 …
In this segment we start a new topic. We will talk about the covariance of two random variables, which gives us useful information about the dependencies between these two random variables. Let us motivate the concept by looking first at a special case. Suppose that X and Y have zero means and that they there …
In this brief segment, we will discuss an important application of the convolution formula. Suppose that X is a normal random variable with a given mean and variance. So that the PDF of X takes this form. And similarly, Y is normal with a given mean and variance. So its PDF takes this form. We …
Probability – L12.4 The Sum of Independent Normal Random Variables Read More »
We now develop a methodology for finding the PDF of the sum of two independent random variables, when these random variables are continuous with known PDFs. So in that case, Z will also be continuous and so will have a PDF. The development is quite analogous to the one for the discrete case. And in …
Probability – L12.3 The Sum of Independent Continuous Random Variables Read More »
We work with discrete random variables as a warm up. And later, we will consider the case of continuous random variables. So suppose that we know the PMFs of X and Y and that we want to compute the probability that the sum is equal to 3. It always helps to have a picture. The …
Probability – L12.2 The Sum of Independent Discrete Random Variables Read More »
This lecture consists of two parts that deal with two rather different topics. In the first part, we look into an important special case of a derived distribution problem. We start with two independent random variables with known distributions and wish to find the distribution of their sum. We will see that for either the …