Discrete Random Variables and Probability Distributions

What is a discrete random variable?

A random variable is a numeric function of the outcome of a random experiment. A discrete random variable takes values in a countable set — typically the integers $0, 1, 2, \ldots$.

Its behavior is summarized by a probability mass function (PMF) $p(k) = P(X = k)$, which must satisfy:

• $p(k) \ge 0$ for every $k$,
• $\sum_k p(k) = 1$.

Step-by-step construction

Setup: Flip a fair coin 3 times. Each of the $2^3 = 8$ ordered outcomes is equally likely (probability $1/8$). Let $X$ count the number of heads.

Step 1 — Enumerate all 8 outcomes and their $X$ values:

• TTT → 0
• HTT, THT, TTH → 1
• HHT, HTH, THH → 2
• HHH → 3

Step 2 — Count favorable outcomes for each $X$ value:

• $X = 0$: 1 outcome.
• $X = 1$: 3 outcomes.
• $X = 2$: 3 outcomes.
• $X = 3$: 1 outcome.

Step 3 — Build the PMF (each entry is count divided by 8):

• $P(X = 0) = 1/8$
• $P(X = 1) = 3/8$
• $P(X = 2) = 3/8$
• $P(X = 3) = 1/8$

Step 4 — Check the total: $1/8 + 3/8 + 3/8 + 1/8 = 8/8 = 1$ ✓.

Summary statistics

Expected value (mean): $E(X) = \sum_k k \cdot P(X = k) = 0 \cdot \tfrac{1}{8} + 1 \cdot \tfrac{3}{8} + 2 \cdot \tfrac{3}{8} + 3 \cdot \tfrac{1}{8} = \tfrac{12}{8} = 1.5$

Variance: $\text{Var}(X) = E(X^2) - (E(X))^2 = 3 - 2.25 = 0.75$

where $E(X^2) = 0 + 3/8 + 12/8 + 9/8 = 24/8 = 3$.

Standard deviation: $\sigma = \sqrt{0.75} \approx 0.866$.

Recognizing the pattern

The counts $1, 3, 3, 1$ are row 3 of Pascal's triangle. This is no coincidence: $X$ is Binomial($n = 3, p = 1/2$), and $P(X = k) = \binom{3}{k} (1/2)^3$.

Cumulative distribution function (CDF)

$F(k) = P(X \le k)$ sums the PMF from the left:

• $F(0) = 1/8$
• $F(1) = 4/8$
• $F(2) = 7/8$
• $F(3) = 8/8 = 1$

A discrete CDF is a step function jumping at each possible value.

Common mistakes

• Forgetting to enumerate ordered outcomes. HTT, THT, TTH are three different outcomes that all give $X = 1$; collapsing them under-counts.
• Confusing PMF with CDF. PMF gives point probabilities; CDF accumulates from the left.
• Assuming probabilities must be fractions with nice denominators. Any nonnegative numbers that sum to 1 form a valid PMF.

Try it in the visualization

Adjust $n$ (number of flips) and $p$ (heads probability) sliders. The sample-space grid highlights each outcome, and the bar chart of $P(X = k)$ reshapes to match. Toggle to show the CDF as a cumulative line.