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 »
In this lecture, we will deal with a single topic. How to find the distribution, that is, the PMF or PDF of a random variable that is defined as a function of one or more other random variables with known distributions. Why is this useful? Quite often, we construct a model by first defining some …
We now look at an application of the Bayes rule that’s involves a continuous unknown random variable, which we try to estimate based on a discrete measurement. Our model will be as follows. We observe the discrete random variable K, which is Bernoulli, so it can take two values, 1 or 0. And it takes …
Probability – L10.11 Inference of the Bias of a Coin Read More »
We will now use the Bayes rule in an important application that involves a discrete unknown random variable and a continuous measurement. Our discrete unknown random variable will be one that takes the values plus or minus 1 with equal probability. And the measurement will be another random variable, Y, which is equal to the …
Probability – L10.10 Detection of a Binary Signal Read More »
We have seen two versions of the Bayes rule– one involving two discrete random variables, and another that involves two continuous random variables. But there are many situations in real life when one has to deal simultaneously with discrete and continuous random variables. For example, you may want to recover a discrete digital signal that …
If you remember our discussion from a long time ago, we said that much of this class consists of variations of a few basic skills and ideas, one of which is the Bayes rule, the foundation of inference. So let’s look here at the Bayes rule again and its different incarnations. In a discrete setting …