2.7 Moving Between PDF and CDF
The book defines pmf and cmf first as a way of developing intuition and a way of reasoning about these concepts. It then moves to defining continuous density functions, which is many ways are easier to work with although they lack the means of reasoning about them intuitively. Continuous distributions are defined in the book, and more generally, in terms of the cdf, which is the cumulative distribution function. There are technical reasons for this choice of definition, some of which are signed in the footnotes on the page where the book presents it.
More importantly for this course, in Definition 1.2.15 the book defines the relationship between cdf and pdf in the following way:
Definition 2.4 (Probability Density Function (PDF)) For a continuous random variable \(X\) with CDF \(F\), the probability density function of \(X\) is
\[ f(x) = \left. \frac{d F(u)}{du} \right|_{u=x}, \forall x \in \mathbb{R}. \]
- How does this definition, which relates pdf and cdf by a means of differentiation and integration, fit with the ideas that we just developed in the context of walking to and from campus?
Example 2.3 (Working with a continuous pdf and cdf) Suppose that you learn than a particular random variable, \(X\) has the following function that describes its pdf, \(f_{x}(x) = \frac{1}{10}x\). Also, suppose that you know that the smallest value that is possible for this random variable to obtain is 0.
- What is the CDF of \(X\)?
- What is the maximum possible value that \(x\) can obtain? How did you develop this answer, using the Kolmogorov axioms of probability?
- What is the cumulative probability of an outcome up to 0.5?
- What is the probability of an outcome between 0.25 and 0.75? Produce an answer to this in two ways:
- Using the \(pdf\)
- Using the \(cdf\)