Calculating a Cumulative Distribution Function (CDF)

What this lecture covers

This recitation problem practices two closely linked skills for continuous random variables: finding an unknown normalizing constant in a probability density function, and integrating that density to build its cumulative distribution function (CDF). It builds on the definitions of PDF and CDF introduced earlier in the course.

You'll see how the requirement that a PDF integrates to 1 pins down an unknown constant, and how to carefully split the real line into ranges (where the density is zero, and where it isn't) to write out a complete, piecewise CDF formula.

Key ideas

• A PDF must integrate to 1: setting integral of f(z) dz over the density's support equal to 1 is how you solve for an unknown constant in the density formula.
• Only the support matters for integration: since the density is zero outside its given range, the normalization integral only needs to be taken over that range.
• The CDF accumulates probability from the left: F(z) = integral from -infinity to z of f(y) dy represents P(Z <= z).
• Keeping track of ranges is the hard part: a CDF built from a piecewise PDF needs separate formulas for where the density is zero (CDF is 0), where it's fully accumulated (CDF is 1), and the range in between (CDF is the integral of the density).
• The CDF's lower and upper bounds: below the support of the density the CDF is 0, and above it the CDF is 1, since all probability mass has been accumulated.
• Verifying the result: once you have gamma, you can substitute it back to fully specify the PDF before computing the CDF.

Before you watch

• Know the definitions of a probability density function and a cumulative distribution function for continuous random variables.
• Be comfortable with basic definite integration, including integrating polynomials.
• This drill assumes you've seen the introductory continuous random variables lecture in this course.

Check your understanding

1. Why must you find gamma before you can compute the CDF?
2. Why does the CDF equal 1 for all z greater than the upper bound of the PDF's support?
3. How does the number of ranges needed in a CDF formula relate to the shape of the underlying PDF?
4. What would go wrong if you forgot to restrict the normalization integral to the PDF's support?