Introduction to Random Variables

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

A random variable maps outcomes to numerical values, discrete (countable) or continuous (any value in a range).

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.