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 …
Simulation is an important tool in the analysis of probabilistic phenomena. For example, suppose that X, Y, and Z are independent random variables, and you’re interested in the statistical properties of this random variable. Perhaps you can find the distribution of this random variable by solving a derived distribution problem, but sometimes this is impossible. …
In all of the examples that we have seen so far, we have calculated the distribution of a random variable, Y, which is defined as a function of another random variable, X. What about the case where we define a random variable, Z, as a function of multiple random variables? For example, here is the …
Probability – L11.9 The PDF of a Function of Multiple Random Variables Read More »
All of our examples so far have involved functions, g of x, that are monotonic in X, at least over the range of x’s of interest. Let us now look at an example that involves a non-monotonic function. We’re going to consider the square function, which has the shape shown in this diagram. And we’ll …
The formula that we just derived for the monotonic case has a nice intuitive explanation that we will develop now. Suppose that g is a monotonic function of x and that it’s monotonically increasing. Let us fix a particular x and a corresponding y so that the two of them are related as follows– y …
Probability – L11.7 The Intuition for the Monotonic Case Read More »