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 »
We have already worked through some examples in which X was a random variable with a given PDF, and we considered the problem of finding the PDF of Y for the case where Y was the function x cubed or the function of the form a/X. What both of these examples have in common is …
In this important segment, we will develop a method for finding the PDF of a general function of a continuous random variable, a function g of X, which, in general, could be nonlinear. The method is very general and involves two steps. The first step is to find the CDF of Y. And then the …
Probability – L11.5 The PDF of a General Function Read More »
Let us now consider an application of what we have done so far. Let X be a normal random variable with given mean and variance. This means that the PDF of X takes the familiar form. We consider random variable Y, which is a linear function of X. And to avoid trivialities, we assume that …
Probability – L11.4 A Linear Function of a Normal Random Variable Read More »
We now move to the case of continuous random variables. We will start with a special case where we want to find the PDF of a linear function of a continuous random variable. We will start by considering a simple example, and study it using an intuitive argument. And afterwards, we will justify our conclusions …
Probability – L11.3 A Linear Function of a Continuous Random Variable Read More »
As a warm-up towards finding the distribution of the function of random variables, let us start by considering the discrete case. So let X be a discrete random variable and let Y be defined as a given function of X. We know the PMF of X and wish to find the PMF of Y. Here’s …
Probability – L11.2 The PMF of a Function of a Discrete Random Variable Read More »