Introduction to Random Variables

A random variable $X$ is a numerical quantity whose value depends on the outcome of a random experiment.

Discrete random variables can take only countable, distinct values — usually integers.

• Number of heads in 10 coin flips: $X \in \{0, 1, 2, \dots, 10\}$
• Sum of two dice: $X \in \{2, 3, 4, \dots, 12\}$
• Number of defective items in a batch: $X \in \{0, 1, 2, \dots, n\}$

Continuous random variables can take any value in an interval — uncountably many possibilities.

• Height of a student: $X \in (0, 3)$ metres
• Time until the next bus: $X \in (0, \infty)$ minutes
• Volume of liquid in a bottle: $X \in (0, 1000)$ mL

The critical difference: For a discrete variable, $P(X = x)$ can be positive. For a continuous variable, $P(X = x) = 0$ for any specific value — we can only talk about $P(a < X < b)$.

What about shoe size? Shoe size takes specific countable values (6, 6.5, 7, 7.5, …), so it is technically discrete — though with many possible values, it is sometimes treated as continuous in practice.

A random variable $X$ maps random outcomes to numerical values; Discrete: countable values — $P(X = x)$ can be positive; use a probability function

Pause — copy the key distinction: discrete random variables have $P(X = x) > 0$ for countable values (e.g., number of heads); continuous random variables have $P(X = x) = 0$ for any specific value — only $P(a < X < b)$ is non-zero — into your book.

We just saw that a discrete random variable assigns positive probability to each countable outcome. That raises a question: what rules must those probabilities satisfy — and how do we find an unknown constant $k$ in a PMF? This card answers it → the two axioms $0 \leq p(x) \leq 1$ and $\sum p(x) = 1$, and using the sum condition to solve for $k$.

The probability function (PMF) gives the probability that $X$ takes each possible value: $p(x) = P(X = x)$.

Finding the constant $k$: A biased die has $P(X = x) = kx$ for $x = 1, 2, 3, 4, 5, 6$.

Use $\sum p(x) = 1$: $k(1 + 2 + 3 + 4 + 5 + 6) = 1$, so $21k = 1$, giving $k = \dfrac{1}{21}$.

Verification: $\dfrac{1}{21} + \dfrac{2}{21} + \dots + \dfrac{6}{21} = \dfrac{21}{21} = 1$ ✓

Worked examples · 3 in a row, reveal as you go

Probability function: $p(x) = P(X = x)$; every value in $[0, 1]$ and all values sum to 1; Finding $k$: set $\sum p(x) = 1$ and solve

Pause — copy the PMF definition $p(x) = P(X = x)$, the two validity conditions ($0 \leq p(x) \leq 1$ and $\sum p(x) = 1$), and the method for finding $k$ (set sum equal to 1 and solve) into your book.

We just saw the PMF gives individual-value probabilities with $\sum p(x) = 1$. That raises a question: exams often ask for $P(X \leq 4)$ or $P(2 \leq X \leq 5)$ — how do we accumulate probabilities and compute these efficiently? This card answers it → the CDF $F(x) = P(X \leq x)$, which accumulates probabilities as a staircase and lets us compute interval probabilities via $F(b) - F(a)$.

Properties of $F(x)$:

• $0 \leq F(x) \leq 1$ for all $x$
• $F(x)$ is non-decreasing (never goes down)
• $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to +\infty} F(x) = 1$
• $P(a < X \leq b) = F(b) - F(a)$

Example — biased die with $k = \frac{1}{21}$:

$F(4) = P(X \leq 4) = \dfrac{10}{21} \approx 0.476$.

Continuous variables and the CDF: For continuous variables, $P(X = x) = 0$ for every $x$. Instead, probability comes from a probability density function (PDF) $f(x)$, where:

• $f(x) \geq 0$ and $\int_{-\infty}^{+\infty} f(x)\,dx = 1$
• $P(a < X < b) = \int_a^b f(x)\,dx$ (area under the curve)
• For continuous variables: $P(a < X < b) = P(a \leq X \leq b)$ since endpoints contribute zero

The CDF for continuous variables is a smooth S-curve (no jumps), unlike the step function for discrete variables.

$F(x) = P(X \leq x)$ — accumulates probability from left up to $x$; Discrete CDF: staircase (jumps at each value of $X$)

Pause — copy the CDF definition $F(x) = P(X \leq x) = \sum_{t \leq x} p(t)$, the interval formula $P(a < X \leq b) = F(b) - F(a)$, and the staircase shape (discrete: jumps at each value; continuous: smooth S-curve) into your book.