Ferkans — Interactive Telecom Tutor

From Events to Numbers

Up to this point, we have worked with events — subsets of a sample space $\Omega$ . But in engineering and science, we almost always care about numerical outcomes: the number of bit errors in a block, the signal power at a receiver, the delay until the next packet arrives. A random variable is the formal device that assigns a number to each outcome, allowing us to bring the full machinery of calculus and algebra to bear on probability problems. The key insight, as Caire emphasizes, is that once we have the distribution of a random variable, "we can forget about the underlying probability space and work entirely with numbers."

Definition:
Random Variable

Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ , a random variable is a function $X : \Omega \to \mathbb{R}$ that is $\mathcal{F}$ -measurable, meaning that for every $x \in \mathbb{R}$ ,

$\{\omega \in \Omega : X(\omega) \leq x\} \in \mathcal{F}.$

This condition ensures that $\mathbb{P}(X \leq x)$ is well-defined for every $x$ .

The measurability condition is not merely a technicality. It guarantees that we can compute the probability of every event of the form $\{X \leq x\}$ , which is sufficient to determine the entire probability law of $X$ .

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Definition:
Discrete Random Variable

A random variable $X$ is discrete if it takes values in a countable subset $\mathcal{X} = \{x_1, x_2, \ldots\} \subset \mathbb{R}$ . The function

$P(x_i) \triangleq \mathbb{P}(X = x_i), \quad i = 1, 2, \ldots$

is called the probability mass function (PMF) of $X$ . A valid PMF satisfies:

$P(x_i) \geq 0$ for all $i$ , and
$\sum_{i} P(x_i) = 1$ .

Definition:
Cumulative Distribution Function (Discrete Case)

The cumulative distribution function (CDF) of a random variable $X$ is

$F(x) \triangleq \mathbb{P}(X \leq x), \quad x \in \mathbb{R}.$

For a discrete RV with PMF $P$ , the CDF is a right-continuous staircase:

$F(x) = \sum_{x_i \leq x} P(x_i).$

The CDF is always non-decreasing, right-continuous, with $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to +\infty} F(x) = 1$ . For a discrete RV, the jump at each point $x_i$ equals $P(x_i) = \mathbb{P}(X = x_i)$ .

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Theorem: Properties of the CDF

Let $F(x)$ be the CDF of a random variable $X$ . Then:

$F$ is non-decreasing.
$F$ is right-continuous: $\lim_{h \downarrow 0} F(x+h) = F(x)$ .
$\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to +\infty} F(x) = 1$ .
$\mathbb{P}(a < X \leq b) = F(b) - F(a)$ .
$\mathbb{P}(X = x) = F(x) - \lim_{h \downarrow 0} F(x - h)$ .

Conversely, any function satisfying (1)--(3) is the CDF of some random variable.

Proof

Non-decreasing

If $a \leq b$ , then $\{X \leq a\} \subseteq \{X \leq b\}$ , so $F(a) = \mathbb{P}(X \leq a) \leq \mathbb{P}(X \leq b) = F(b)$ by monotonicity of probability.

Right-continuity

Define $A_n = \{X \leq x + 1/n\}$ for $n = 1, 2, \ldots$ . Then $A_n \downarrow \{X \leq x\}$ as $n \to \infty$ , and by continuity of probability from above, $F(x + 1/n) = \mathbb{P}(A_n) \to \mathbb{P}(X \leq x) = F(x)$ .

Limits at infinity

The events $\{X \leq n\} \uparrow \Omega$ as $n \to \infty$ , so $F(n) \to \mathbb{P}(\Omega) = 1$ . Similarly $\{X \leq -n\} \downarrow \emptyset$ , giving $F(-n) \to 0$ . $\blacksquare$

Example: The Bernoulli Random Variable

A coin lands heads with probability $p \in (0,1)$ . Define $X = 1$ if heads, $X = 0$ if tails. Find the PMF and CDF of $X$ .

Solution

PMF

$P(0) = 1 - p$ and $P(1) = p$ . The support is $\mathcal{X} = \{0, 1\}$ .

CDF

$F(x) = \begin{cases} 0 & x < 0 \\ 1 - p & 0 \leq x < 1 \\ 1 & x \geq 1 \end{cases}$ $This is a staircase with jumps of size$ 1 - p $at$ x = 0 $and size$ p $at$ x = 1$.

Definition:
Indicator Random Variable

For an event $A \in \mathcal{F}$ , the indicator random variable $\mathbf{1}_A$ is defined by

$\mathbf{1}_A(\omega) = \begin{cases} 1 & \text{if } \omega \in A \\ 0 & \text{if } \omega \notin A \end{cases}$

Notice that $\mathbb{E}[\mathbf{1}_A] = \mathbb{P}(A)$ . The Bernoulli RV is the indicator of the event "heads."

Indicator random variables are a powerful bookkeeping device. Many counting problems become tractable when we express the count as a sum of indicators and exploit linearity of expectation.

Example: Counting Matches via Indicators

In a random permutation of $\{1, 2, \ldots, n\}$ , let $X$ be the number of fixed points (elements that remain in their original position). Find the PMF of $X$ for $n = 3$ .

Solution

Express as a sum of indicators

Define $X_i = \mathbf{1}\{i \text{ is a fixed point}\}$ for $i = 1, \ldots, n$ . Then $X = X_1 + X_2 + \cdots + X_n$ .

Enumerate for $n = 3$

The $3! = 6$ permutations of $\{1, 2, 3\}$ are: $(1,2,3)$ , $(1,3,2)$ , $(2,1,3)$ , $(2,3,1)$ , $(3,1,2)$ , $(3,2,1)$ .

The fixed-point counts are: $3, 1, 1, 0, 0, 1$ .

So $P(0) = 2/6 = 1/3$ , $P(1) = 3/6 = 1/2$ , $P(2) = 0$ (impossible — if two are fixed, the third must be too), $P(3) = 1/6$ .

PMF and CDF of a Discrete Distribution

Explore the PMF and CDF of a Bernoulli or binomial distribution. Observe how the CDF is a right-continuous staircase with jumps matching the PMF values.

Parameters

Distribution

p

0.5

n

(binomial only)10

Quick Check

Which of the following functions is a valid PMF on $\mathcal{X} = \{1, 2, 3\}$ ?

$p(1) = 0.3,\ p(2) = 0.3,\ p(3) = 0.3$

$p(1) = 0.5,\ p(2) = 0.3,\ p(3) = 0.2$

$p(1) = 0.5,\ p(2) = -0.1,\ p(3) = 0.6$

$p(1) = 1.2,\ p(2) = -0.1,\ p(3) = -0.1$

Correction:

p(1) = 0.5,\ p(2) = 0.3,\ p(3) = 0.2

All values are non-negative and $0.5 + 0.3 + 0.2 = 1$ .

Common Mistake: PMF Is a Probability, PDF Is a Density

Mistake:

Treating $P(x)$ and a probability density function $f_X(x)$ as the same kind of object. Students often assume $f_X(x) \leq 1$ .

Correction:

For a discrete RV, $P(x) = \mathbb{P}(X = x)$ is an actual probability, so $0 \leq P(x) \leq 1$ . For a continuous RV, $f_X(x)$ is a density and can exceed 1 — only $\int f_X(x)\,dx$ must equal 1.

Random Variable

A measurable function $X : \Omega \to \mathbb{R}$ from the sample space to the reals. "Random" refers to the randomness in $\omega$ ; the function $X$ itself is deterministic.

Probability Mass Function (PMF)

For a discrete random variable $X$ with support $\mathcal{X}$ , $P(x) = \mathbb{P}(X = x)$ for $x \in \mathcal{X}$ .

Cumulative Distribution Function (CDF)

$F(x) = \mathbb{P}(X \leq x)$ . A non-decreasing, right-continuous function from $\mathbb{R}$ to $[0, 1]$ .

Historical Note: The Birth of Random Variables

1933

The concept of a random variable as a measurable function was formalized by Andrey Kolmogorov in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability). Before Kolmogorov, probabilists like Laplace and Chebyshev worked with "quantities depending on chance" without a rigorous definition. Kolmogorov's measure-theoretic framework placed probability on the same axiomatic footing as the rest of modern mathematics.

Key Takeaway

A random variable is a function, not a number. The randomness comes from the input $\omega \in \Omega$ , not from the function itself. Once we have the PMF (or CDF), we can forget about the underlying sample space and work entirely with numbers.

Definition and Probability Mass Function