August 2025

In this lecture, we continue our discussion of continuous random variables. We will start by bringing conditioning into the picture and discussing how the PDF of a continuous random variable changes when we are told that a certain event has occurred. We will take the occasion to develop counterparts of some of the tools that …

Probability – L09.1 Lecture Overview – Conditioning on an event; Multiple RVs Read More »

We have claimed that normal random variables are very important, and therefore we would like to be able to calculate probabilities associated with them. For example, given a normal random variable, what is the probability that it takes a value less than 5? Unfortunately, there are no closed form expressions that can help us with …

Probability – L08.9 Calculation of Normal Probabilities Read More »

We now introduce normal random variables, which are also often called Gaussian random variables. Normal random variables are perhaps the most important ones in probability theory. They play a key role in the theory of the subject, as we will see later in this class in the context of the central limit theorem. They’re also …

Probability – L08.8 Normal Random Variables Read More »

We have seen that several properties, such as, for example, linearity of expectations, are common for discrete and continuous random variables. For this reason, it would be nice to have a way of talking about the distribution of all kinds of random variables without having to keep making a distinction between the different types– discrete …

Probability – L08.7 Cumulative Distribution Functions Read More »

We now introduce a new of random variable, the exponential random variable. It has a probability density function that is determined by a single parameter lambda, which is a positive number. And the form of the PDF is as shown here. Note that the PDF is equal to 0 when x is negative, which means …

Probability – L08.6 Exponential Random Variables Read More »

As an example of a mean-variance calculation, we will now consider the continuous uniform random variable which we have introduced a little earlier. This is the continuous analog of the discrete uniform, for which we have already seen formulas for the corresponding mean and variance. So let us now calculate the mean or expected value …

Probability – L08.5 Mean & Variance of the Uniform Read More »

We now start with our agenda of developing continuous counterparts of everything we have done for discrete random variables. Let us look at the concept of expectation. In the discrete case, we have defined expectation as a weighted average of the values X of the random variable, weighted according to their corresponding probabilities. In the …

Probability – L08.4 Means & Variances Read More »

Let us now give an example of a continuous random variable– the uniform random variable. It is patterned after the discrete random variable. Similar to the discrete case, we will have a range of possible values. In the discrete case, these values would be the integers between a and b. In the continuous case, any …

Probability – L08.3 Uniform & Piecewise Constant PDFs Read More »

In this segment, we introduce the concept of continuous random variables and their characterization in terms of probability density functions, or PDFs for short. Let us first go back to discrete random variables. A discrete random variable takes values in a discrete set. There is a total of one unit of probability assigned to the …

Probability – L08.2 Probability Density Functions Read More »

In this lecture, we start our discussion of continuous random variables. We will focus on the case of a single continuous random variable, and we’ll describe its distribution using a so-called probability density function, an object that will replace the PMFs from the discrete case. We will then proceed to define the expectation and the …

Probability – L08.1 Lecture Overview – Continuous RV and PDFs Read More »