Lecture 12

Cumulative Distribution Functions


Grace Tompkins

Last modified — 23 Oct 2026

Learning Outcomes

By the end of this lecture, students are anticipated to be able to:

  • Define a CDF
  • Use a CDF to calculate cumulative probabilities
  • Convert between PDFs and CDFs
  • Find Mixture Distributions

1 Cumulative Distribution Functions (CDFs)

Cumulative Distribution Functions (CDFs)

The cumulative distribution function of a random variable \(X\) is \[F_X(x) \, = \, \mathbb{P}\left( X \le x \right) \, , \qquad \mbox{ for } x \in {\mathbb{R}}\]

Let \(X\) be a random variable with CDF \(F_X\). Then:

  1. \(0 \le F_X(x) \le 1\) for all \(x \in {\mathbb{R}}\);
  2. \(F\left( x \right) \leq F(y)\) for all \(x \leq y\);
  3. \(\lim_{a \to -\infty} F_X\left( a \right) =0\),
  4. \(\lim_{a \to +\infty} F_X\left( a \right) =1\)

CDFs for Discrete and Continuous RVs

Discrete RV

\[F_X(x) = \sum_{t \leq x} p_X(t)\]

  • The CDF is a step function and right-continuous.

Continuous RV

\[F_X(x) = \int_{-\infty}^x f_X(t) \mathsf{d}t\]

  • The CDF is continuous (left and right limits are equal).

Continuous CDF Example

Let \(X \sim {\mathrm{Exp}}(\lambda)\), with pdf \[f_X(x) = \lambda e^{-\lambda x} I_{[0,\infty)}(x).\]

Find \(F_{X}\left(x \right)\). Use the CDF to calculate \(\mathbb{P}(x \le 10)\) when \(\lambda = 1/5\).

Continuous CDF Example

Discrete CDF Example

Consider rolling one fair six-sided die, so that \(\Omega = \{1, 2, 3, 4, 5, 6\}\), with \(\mathbb{P}(\omega) = 1/6\) for each \(\omega \in \Omega\) Let \(X\) be the number showing on the die. What is \(F_{X}(x)\)? Use this to find \(\mathbb{P}(X \le 5)\).

CDFs and Distributions

Let \(X\) be any random variable with CDF \(F_X\). Let \(B\) be any subset of \({\mathbb{R}}\). Then \(\mathbb{P}(X \in B)\) can be determined solely from \(F_X\).

  • The CDF contains all the information about the distribution of \(X\).
  • It doesn’t matter whether \(X\) is discrete, continuous, or anything else.

CDFs and Distributions

Let \(X\) be an absolutely continuous random variable with cumulative distribution function \(F_X\). Let

\[ f_X(x) = \frac{d}{dx}F_X(x) = F'(x). \] Then, \(f(x)\) is the density function for \(X\).

Aside: The discrete case involves finding the differences between consecutive CDF values.

CDFs and Distributions

Suppose a continuous RV has CDF:

\[ F_X(x) = \begin{cases} 0 & x < 0\\ x^2 & 0 \le x \le 1\\ 1 & x>1 \end{cases} \]

  1. What is \(\mathbb{P}(X \le 0.75)\)?

  2. Find the PDF \(f_X(x)\).

CDFs and Distributions

CDFs and Distributions

Suppose a discrete RV has CDF:

\[ F_X(x) = \begin{cases} 0 & x < 1 \\ 0.1 & 1 \le x < 2 \\ 0.35 & 2 \le x < 3 \\ 0.7 & 3 \le x < 4 \\ 1 & x \ge 4 \end{cases} \]

  1. What is \(\mathbb{P}(X \le 3.5)\)?

  2. Find the PDF \(f_X(x)\).

CDFs and Distributions

Mixture Distributions

Let \(X_1, X_2,\dots\) be a random variables with CDFs \(F_{X_1}, F_{X_2}, \dots\).

For any constants \(p_i\) such that \(p_i \ge 0\) and \(\sum_{i = 1}^k p_i = 1\), \(F_G(x) = \sum_{i = 1}^k p_i F_{X_i}(x)\) is the CDF of the mixture of \(F_{X_i}\).

Mixture Distributions

Suppose a bag contains two coins. You choose one at random and flip it once. Let \(X\) be the number of heads.

Coin A is a fair coin, with \(\mathbb{P}(\text{flip Heads}) = 0.5\) Coin B is a loaded coin, with \(\mathbb{P}(\text{flip Heads}) = 0.9\)

Write out the CDF for the number of heads flipped.

Mixture Distributions

Mixture Distributions

Consider the scores of a midterm. Suppose that there are three types of students, modeled as follows:

  1. Poorly prepared students who did not study much (30%): \(X_1 \sim \mathcal{N}(\mu = 50, \sigma = 10)\).
  2. Well prepared students who studied a lot (60%): \(X_2 \sim \mathcal{N}(\mu = 80, \sigma = 8)\).
  3. Students who didn’t take the exam at all (10%): \(X_3 = 0\) with probability 1.

Find the CDF of the overall score distribution \(X\). Write your answer in terms of the standard normal CDF \(\Phi(\cdot)\).

Hint: If \(Y \sim \mathcal{N}(\mu, \sigma)\), then \(F_Y(y) = \Phi\left( \frac{y - \mu}{\sigma} \right)\).

Mixture Distributions

Upcoming Assessments:

  • Checkpoint 4 (Sections 2.5 - 2.6: Continuous RV and CDFs) is on Friday Oct 30th

  • Checkpoint 5 (Sections 2.7 - 2.8: Joint, Marginal, and Conditional Distributions) is on Friday Nov 6th