Category: Continuous Random Variables

Roughly speaking, the median is a value splitting a dataset into two equal parts. When dealing with continuous random variables, we find that the median is a value mX such that Geometrically, the median splits the PDF in two parts with an area equal to 0.5. In descriptive statistics, the median can be regarded as a specific…

The descriptive statistics, we have introduced concepts like mode, median, and percentiles. We have also remarked that some concepts, in particular the percentiles, are somewhat shaky in the sense that there are slightly different definitions and ways of calculating them using observed data. In this section we examine probabilistic counterparts of these concepts, and how…

Given a continuous random variable X and its PDF fX(x), its expected value is defined as follows: Quite often, we use the short-hand notation μX = E[X]. Again, this is straightforward extension of the discrete case, where E[X] ≡ ∑i xipX(xi). Example 7.2 As an illustration, let us consider the expected value of a uniform random variable on [a, b]. Symmetry suggests that…

A full characterization of discrete random variables can be given in terms of PMF or CDF. They are related, as the CDF can be obtained from the PMF by summing, and we can go the other way around by taking differences. For the reasons we have mentioned, in the continuous case the role of the…

The most natural way to characterize a discrete distribution is by its PMF, which can be depicted as a set of bars whose height is the probability of each value. What happens when we consider a random variable that may take any real value on an interval? A starting point to build intuition is getting…

We have gained the essential intuition about random variables in the discrete setting. There, we introduced ways to characterize the distribution of a random variable by its PMF and CDF, as well as its expected value and variance. Now we move on to the more challenging case of a continuous random variable. There are several…