Last updated: February 12, 2024 - Added sections on chemical kinetics and simulated annealing

Discrete Random Variables and Mixed Distributions

We discussed continuous random variables in class and characterized their distributions with CDFs and PDFs. Discrete random variables are quantities that can take on a discrete or countable number of values. Typical discrete random variables include counts, such as the number of times a detector is hit within a fixed time interval, or Bernoulli trials (i.e., biased coin flips that model one of two possible outcomes, such as success/failure or heads/tails).

Example (Bernoulli Trial)

Flip a coin with bias $p$ and let $X$ be the discrete random variable, called a Bernoulli RV, defined as:

$$ X = \begin{cases}0 & \text{ if heads } \\ 1 & \text{ if tails}\end{cases} $$

Note that nothing is sacred about the two values $0$ and $1$ above. We could define another Bernoulli variable as:

Example (Bernoulli, again)

Flip a coin with bias $p$ and let $Y$ be the discrete random variable, also a Bernoulli RV, defined as:

$$ Y = \begin{cases}-1 & \text{ if heads } \\ 1 & \text{ if tails}\end{cases} $$

Like continuous RVs, we can define a CDF as

$$ F_X(x) = \mathbb P[X \leq x] $$

though, unlike continuous RVs, CDFs of discrete RVs are step-functions:

Example (Bernoulli RV)

Let $Y$ be the Bernoulli RV defined above. Then, the CDF for $Y$ is given by the step function: