Ferkans — Interactive Telecom Tutor

Why Random Variables?

A probability space $(\Omega, \mathcal{F}, P)$ gives us the formal machinery to talk about random experiments, but the sample space $\Omega$ can be unwieldy --- outcomes might be waveforms, sequences of channel states, or abstract protocol events. To compute, we need numbers.

A random variable is the bridge: it maps every outcome $\omega \in \Omega$ to a real number, converting abstract randomness into something we can integrate, differentiate, and optimise.

In wireless communications the idea is immediate. The received baseband sample at a single antenna is

$r = s + n,$

where $s$ is the transmitted symbol (deterministic, once the codebook is fixed) and $n$ is additive noise --- a Gaussian random variable. Understanding the distribution of $n$ lets us compute bit-error rates, design detectors, and prove capacity theorems. Every subsequent chapter builds on this section's definitions.

Definition:
Random Variable

Let $(\Omega, \mathcal{F}, P)$ be a probability space. A random variable is a measurable function

$X : \Omega \to \mathbb{R},$

meaning that for every Borel set $B \subseteq \mathbb{R}$ the pre-image $X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\}$ belongs to the $\sigma$ -algebra $\mathcal{F}$ .

In particular, the set $\{X \le x\} \triangleq \{\omega \in \Omega : X(\omega) \le x\}$ is an event (element of $\mathcal{F}$ ) for every $x \in \mathbb{R}$ , so $P(X \le x)$ is well-defined.

The measurability condition is automatically satisfied for every function encountered in engineering practice. Its role is to prevent pathological constructions that would make probabilities undefined.

We write " $X$ " (uppercase italic) for a random variable and " $x$ " (lowercase italic) for a particular realisation.

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Definition:
Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable $X$ is the function $F_X : \mathbb{R} \to [0, 1]$ defined by

$F_X(x) = P(X \le x), \qquad x \in \mathbb{R}.$

Properties. Every CDF satisfies:

Right-continuity: $F_X$ is right-continuous at every point, i.e., $\lim_{h \downarrow 0} F_X(x + h) = F_X(x)$ .
Monotonicity: $F_X$ is non-decreasing: $x_1 < x_2 \;\Longrightarrow\; F_X(x_1) \le F_X(x_2)$ .
Boundary limits: $\displaystyle\lim_{x \to -\infty} F_X(x) = 0$ and $\displaystyle\lim_{x \to +\infty} F_X(x) = 1$ .

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

The CDF completely characterises the distribution of $X$ . Two random variables with the same CDF are said to be equal in distribution, written $X \stackrel{d}{=} Y$ .

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Definition:
Probability Density Function (PDF)

A random variable $X$ is called continuous (or, more precisely, absolutely continuous) if there exists a non-negative function $f_X : \mathbb{R} \to [0, \infty)$ such that

$F_X(x) = \int_{-\infty}^{x} f_X(t)\,dt, \qquad \forall\, x \in \mathbb{R}.$

The function $f_X$ is called the probability density function (PDF) of $X$ . At every point where $F_X$ is differentiable,

$f_X(x) = \frac{dF_X(x)}{dx}.$

Properties.

$f_X(x) \ge 0$ for all $x$ .
$\displaystyle\int_{-\infty}^{\infty} f_X(x)\,dx = 1$ .
$\displaystyle P(a < X \le b) = \int_a^b f_X(x)\,dx$ .

The PDF is not a probability: $f_X(x)$ can exceed $1$ . Only when integrated over an interval does it yield a probability.

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Definition:
Probability Mass Function (PMF)

A random variable $X$ is discrete if it takes values in a countable set $\mathcal{X} = \{x_1, x_2, \dots\}$ . Its probability mass function (PMF) is

$p_X(x) = P(X = x), \qquad x \in \mathcal{X}.$

Properties.

$p_X(x) \ge 0$ for all $x \in \mathcal{X}$ .
$\displaystyle\sum_{x \in \mathcal{X}} p_X(x) = 1$ .

The CDF of a discrete random variable is a staircase function with jumps of size $p_X(x_k)$ at each point $x_k \in \mathcal{X}$ .

In digital communications, discrete random variables arise naturally: the transmitted bit $b \in \{0, 1\}$ , the number of errors in a codeword, and the number of packet arrivals in a time slot are all discrete.

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Definition:
Expectation (Mean)

The expectation (or mean) of a random variable $X$ is defined as

$E[X] = \int_{-\infty}^{\infty} x\,f_X(x)\,dx$

for continuous $X$ (provided the integral converges absolutely), and

$E[X] = \sum_{x \in \mathcal{X}} x\,p_X(x)$

for discrete $X$ . We also write $\mu_X = E[X]$ .

LOTUS (Law of the Unconscious Statistician). For any (measurable) function $g : \mathbb{R} \to \mathbb{R}$ ,

$E[g(X)] = \int_{-\infty}^{\infty} g(x)\,f_X(x)\,dx$

without needing to derive the distribution of $g(X)$ first.

Linearity of expectation. For any random variables $X$ and $Y$ (not necessarily independent) and constants $a, b \in \mathbb{R}$ ,

$E[aX + bY] = a\,E[X] + b\,E[Y].$

LOTUS is one of the most practically useful results in probability. For example, the average received power is $E[|r|^2]$ , which LOTUS lets us compute directly from the distribution of $r$ without first finding the distribution of $|r|^2$ .

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Definition:
Variance and Standard Deviation

The variance of a random variable $X$ is

$\mathrm{Var}(X) = E\bigl[(X - \mu_X)^2\bigr] = E[X^2] - \bigl(E[X]\bigr)^2,$

where $\mu_X = E[X]$ . We write $\sigma_X^2 = \mathrm{Var}(X)$ .

The standard deviation is $\sigma_X = \sqrt{\mathrm{Var}(X)}$ .

Properties.

$\mathrm{Var}(X) \ge 0$ , with equality iff $X = \mu_X$ a.s.
$\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)$ for constants $a, b$ .
If $X$ and $Y$ are independent, then $\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$ .

The identity $\mathrm{Var}(X) = E[X^2] - (E[X])^2$ is the computational workhorse. In noise analysis, if $n$ is zero-mean ( $E[n] = 0$ ), then $\mathrm{Var}(n) = E[n^2]$ , which is the noise power.

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Theorem: Properties of Expectation

Let $X$ and $Y$ be random variables on $(\Omega, \mathcal{F}, P)$ with finite expectations, and let $a, b, c \in \mathbb{R}$ . Then:

Linearity: $E[aX + bY + c] = a\,E[X] + b\,E[Y] + c$ .
Monotonicity: If $X \le Y$ almost surely, then $E[X] \le E[Y]$ .
Triangle inequality: $|E[X]| \le E[|X|]$ .
Product rule for independent RVs: If $X$ and $Y$ are independent, then $E[XY] = E[X]\,E[Y]$ .

Linearity is the single most important property of expectation and holds without any independence assumption. It is the reason we can decompose the mean of a sum of interference terms without knowing their joint distribution.

Monotonicity formalises the obvious: if one random variable always dominates another, its average must be at least as large.

The product rule for independent RVs is not a consequence of linearity; it requires the factorisation of the joint PDF.

Show Hint

For linearity, use the definition of expectation and linearity of integration.

For monotonicity, note that $Y - X \ge 0$ a.s. implies $E[Y - X] \ge 0$ .

For the product rule, write $E[XY] = \iint xy\,f_{X,Y}(x,y)\,dx\,dy$ and factor the joint PDF.

Proof

Linearity

For continuous random variables,

$E[aX + bY + c] = \iint (ax + by + c)\,f_{X,Y}(x,y)\,dx\,dy.$

Splitting the integral by linearity of integration:

$= a\!\iint x\,f_{X,Y}(x,y)\,dx\,dy + b\!\iint y\,f_{X,Y}(x,y)\,dx\,dy + c\!\iint f_{X,Y}(x,y)\,dx\,dy.$

The first double integral equals $E[X]$ (integrate out $y$ to get the marginal), the second equals $E[Y]$ , and the third equals $1$ . Hence $E[aX + bY + c] = a\,E[X] + b\,E[Y] + c$ .

Monotonicity

Define $Z = Y - X$ . Then $Z \ge 0$ a.s., so $f_Z(z) = 0$ for $z < 0$ . Therefore

$E[Z] = \int_0^{\infty} z\,f_Z(z)\,dz \ge 0,$

and by linearity $E[Y] - E[X] = E[Z] \ge 0$ , i.e., $E[X] \le E[Y]$ . $\blacksquare$

Product rule for independent RVs

If $X$ and $Y$ are independent, their joint PDF factors: $f_{X,Y}(x,y) = f_X(x)\,f_Y(y)$ . Then

$E[XY] = \iint xy\,f_X(x)\,f_Y(y)\,dx\,dy = \left(\int x\,f_X(x)\,dx\right)\!\left(\int y\,f_Y(y)\,dy\right) = E[X]\,E[Y]. \;\blacksquare$

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Definition:
Bernoulli Distribution

A random variable $X$ has the Bernoulli distribution with parameter $p \in [0, 1]$ , written $X \sim \mathrm{Bernoulli}(p)$ , if

$p_X(x) = \begin{cases} 1 - p, & x = 0, \\ p, & x = 1, \\ 0, & \text{otherwise}. \end{cases}$

Mean: $E[X] = p$ .

Variance: $\mathrm{Var}(X) = p(1 - p)$ .

The Bernoulli distribution is the natural model for a single bit decision. In a binary symmetric channel with crossover probability $p$ , the error indicator $e \sim \mathrm{Bernoulli}(p)$ takes the value $1$ when the received bit differs from the transmitted bit.

Definition:
Binomial Distribution

A random variable $X$ has the binomial distribution with parameters $n \in \mathbb{N}$ and $p \in [0,1]$ , written $X \sim \mathrm{Binomial}(n, p)$ , if

$p_X(k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad k = 0, 1, \dots, n.$

Mean: $E[X] = np$ .

Variance: $\mathrm{Var}(X) = np(1 - p)$ .

If a block of $n$ coded bits is sent through a BSC with crossover probability $p$ , and errors are independent across bits, then the total number of bit errors $K \sim \mathrm{Binomial}(n, p)$ . The binomial distribution thus governs block error analysis for memoryless channels.

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Definition:
Poisson Distribution

A random variable $X$ has the Poisson distribution with rate parameter $\lambda > 0$ , written $X \sim \mathrm{Poisson}(\lambda)$ , if

$p_X(k) = \frac{\lambda^k e^{-\lambda}}{k!}, \qquad k = 0, 1, 2, \dots$

Mean: $E[X] = \lambda$ .

Variance: $\mathrm{Var}(X) = \lambda$ .

The Poisson distribution arises as the limit of $\mathrm{Binomial}(n, p)$ when $n \to \infty$ , $p \to 0$ , and $np \to \lambda$ . In telecommunications it models the number of packet arrivals in a time slot, the number of interferers in a stochastic geometry model, and the number of photons detected in an optical receiver.

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Definition:
Continuous Uniform Distribution

A random variable $X$ has the uniform distribution on $[a, b]$ , written $X \sim \mathrm{Uniform}(a, b)$ , if its PDF is

$f_X(x) = \begin{cases} \dfrac{1}{b - a}, & a \le x \le b, \\[6pt] 0, & \text{otherwise}. \end{cases}$

Mean: $\displaystyle E[X] = \frac{a + b}{2}$ .

Variance: $\displaystyle \mathrm{Var}(X) = \frac{(b-a)^2}{12}$ .

In wireless communications, the carrier phase offset $\Theta \sim \mathrm{Uniform}(0, 2\pi)$ models the fact that the receiver has no a priori knowledge of the absolute phase. This assumption underlies non-coherent detection and is implicit in the derivation of the Rayleigh fading model.

Definition:
Exponential Distribution

A random variable $X$ has the exponential distribution with rate parameter $\lambda > 0$ , written $X \sim \mathrm{Exp}(\lambda)$ , if

$f_X(x) = \begin{cases} \lambda e^{-\lambda x}, & x \ge 0, \\ 0, & x < 0. \end{cases}$

CDF: $F_X(x) = 1 - e^{-\lambda x}$ for $x \ge 0$ .

Mean: $\displaystyle E[X] = \frac{1}{\lambda}$ .

Variance: $\displaystyle \mathrm{Var}(X) = \frac{1}{\lambda^2}$ .

Memoryless property. The exponential is the only continuous distribution with the memoryless property:

$P(X > s + t \mid X > s) = P(X > t), \qquad \forall\, s, t \ge 0.$

The exponential distribution models inter-arrival times in a Poisson process. In teletraffic engineering, the time between successive packet arrivals at a router is often modelled as $\mathrm{Exp}(\lambda)$ . Additionally, in Rayleigh fading the instantaneous SNR $\gamma = |h|^2 \bar{\gamma}$ is exponentially distributed when $h \sim \mathcal{CN}(0, 1)$ .

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Definition:
Gaussian (Normal) Distribution

A random variable $X$ has the Gaussian (or normal) distribution with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$ , written $X \sim \mathcal{N}(\mu, \sigma^2)$ , if its PDF is

$f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right), \qquad x \in \mathbb{R}.$

Mean: $E[X] = \mu$ .

Variance: $\mathrm{Var}(X) = \sigma^2$ .

The standard normal $Z \sim \mathcal{N}(0, 1)$ is obtained by setting $\mu = 0$ , $\sigma^2 = 1$ . Any Gaussian can be standardised: $Z = (X - \mu)/\sigma$ .

The Q-function. The tail probability of the standard normal is

$Q(x) = P(Z > x) = \int_x^{\infty} \frac{1}{\sqrt{2\pi}} e^{-t^2/2}\,dt = \frac{1}{2}\,\mathrm{erfc}\!\left(\frac{x}{\sqrt{2}}\right),$

where $\mathrm{erfc}$ is the complementary error function. This function appears in virtually every error-probability expression in digital communications.

The Gaussian distribution owes its central role to the Central Limit Theorem (Section 2.5): the sum of many independent, identically distributed random variables converges in distribution to a Gaussian, regardless of the individual distributions. Thermal noise, which is the aggregate effect of countless electron vibrations, is therefore accurately modelled as Gaussian.

The complex Gaussian $\mathcal{CN}(\mu, \sigma^2)$ used for baseband noise has independent real and imaginary parts, each $\mathcal{N}(\mu_r, \sigma^2/2)$ and $\mathcal{N}(\mu_i, \sigma^2/2)$ respectively, so that the total power (variance) is $\sigma^2$ .

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Gaussian PDF and CDF Explorer

Visualise how the Gaussian PDF and CDF change as you vary the mean $\mu$ and standard deviation $\sigma$ . Use the dropdown to overlay Rayleigh, Ricean, or Nakagami- $m$ densities --- these fading distributions, derived from the Gaussian in Section 2.3, are previewed here for comparison.

Parameters

Mean

\mu

0

Standard deviation

\sigma

1

Distribution overlay

Show CDF

Example: Bit-Error Rate of BPSK in AWGN

A binary phase-shift keying (BPSK) system transmits $s \in \{+\!\sqrt{E_b},\;-\!\sqrt{E_b}\}$ with equal probability over an AWGN channel. The received signal is

$r = s + n, \qquad n \sim \mathcal{N}\!\left(0, \frac{N_0}{2}\right).$

The receiver decides $\hat{s} = +\!\sqrt{E_b}$ if $r > 0$ and $\hat{s} = -\!\sqrt{E_b}$ if $r \le 0$ . Derive the bit-error rate (BER) as a function of $E_b/N_0$ .

Solution

Condition on transmitted symbol

By symmetry of the constellation and the decision rule,

$P_b = P(\text{error} \mid s = +\!\sqrt{E_b}) = P(\text{error} \mid s = -\!\sqrt{E_b}).$

Condition on $s = +\!\sqrt{E_b}$ :

$P_b = P(r \le 0 \mid s = +\!\sqrt{E_b}) = P\!\left(\sqrt{E_b} + n \le 0\right) = P\!\left(n \le -\sqrt{E_b}\right).$

Standardise the Gaussian

Since $n \sim \mathcal{N}(0, N_0/2)$ , let $Z = n / \sqrt{N_0/2} \sim \mathcal{N}(0, 1)$ . Then

$P_b = P\!\left(Z \le -\frac{\sqrt{E_b}}{\sqrt{N_0/2}}\right) = P\!\left(Z \le -\sqrt{\frac{2E_b}{N_0}}\right).$

Express in terms of the Q-function

Using the symmetry of the standard normal, $P(Z \le -a) = P(Z > a) = Q(a)$ :

$\boxed{P_b = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right).}$

This is the fundamental BPSK BER formula. At $E_b/N_0 = 9.6\;\mathrm{dB}$ (about $9.12$ in linear scale), $P_b \approx 10^{-5}$ .

Equivalent erfc form

Using $Q(x) = \frac{1}{2}\,\mathrm{erfc}(x/\sqrt{2})$ :

$P_b = \frac{1}{2}\,\mathrm{erfc}\!\left(\sqrt{\frac{E_b}{N_0}}\right).$

Both forms are widely used; the Q-function form is standard in information-theoretic analyses, while the erfc form is common in link-budget calculations and simulation scripts.

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The Gaussian Distribution: Shape, Scale, and the 68-95-99.7 Rule

Watch how the Gaussian PDF transforms as

\sigma

sweeps from 0.5 to 3.0, with the 68-95-99.7 rule illustrated as shaded regions.

The Gaussian

\mathcal{N}(\mu, \sigma^2)

concentrates probability near the mean. As

\sigma

grows, the bell curve flattens and spreads, but the area under it remains 1.

Why This Matters: The Q-Function and Error Probability in Digital Communications

The Q-function is the universal currency for error-rate computation in AWGN channels. For any modulation scheme whose conditional error event reduces to a Gaussian tail probability, the BER or symbol-error rate (SER) can be expressed in terms of $Q(\cdot)$ .

Examples:

Modulation	Error probability
BPSK	$P_b = Q\!\left(\sqrt{2E_b/N_0}\right)$
QPSK	$P_s \approx 2\,Q\!\left(\sqrt{2E_s/N_0}\right)$
$M$ -QAM	$P_s \approx 4\!\left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3E_s}{(M-1)N_0}}\right)$

The Q-function decreases faster than any polynomial but slower than a pure exponential. Its asymptotic approximation for large $x$ is

$Q(x) \approx \frac{1}{\sqrt{2\pi}\,x}\,e^{-x^2/2}, \qquad x \gg 1,$

which is useful for high-SNR analysis and diversity-order calculations.

Relation to erfc. The Q-function and complementary error function are related by

$Q(x) = \frac{1}{2}\,\mathrm{erfc}\!\left(\frac{x}{\sqrt{2}}\right), \qquad \mathrm{erfc}(x) = 2\,Q\!\left(x\sqrt{2}\right).$

Most numerical libraries provide $\mathrm{erfc}$ directly, making this conversion essential for simulation code.

See full treatment in Chapter 5، Section 2

Quick Check

If $f_X(x) = c\,x\,e^{-x^2}$ for $x \ge 0$ (and $f_X(x) = 0$ for $x < 0$ ), what value of $c$ makes $f_X$ a valid PDF?

$c = 1$

$c = 2$

$c = 1/2$

$c = 4$

Correction:

c = 2

We need $\int_0^{\infty} c\,x\,e^{-x^2}\,dx = 1$ . Using the substitution $u = x^2$ , $du = 2x\,dx$ : $\int_0^{\infty} x\,e^{-x^2}\,dx = \frac{1}{2}\int_0^{\infty} e^{-u}\,du = \frac{1}{2}$ . Hence $c \cdot \frac{1}{2} = 1$ , giving $c = 2$ . (This is, in fact, the Rayleigh PDF with $\sigma^2 = 1/2$ .)

Quick Check

Let $X \sim \mathcal{N}(3, 4)$ . What is $E[2X + 5]$ ?

$11$

$6$

$13$

$16$

Correction:

11

By linearity, $E[2X + 5] = 2\,E[X] + 5 = 2(3) + 5 = 11$ . Note that the variance $\sigma^2 = 4$ does not affect the mean calculation.

Quick Check

If $\mathrm{Var}(X) = \sigma^2$ , what is $\mathrm{Var}(3X - 7)$ ?

$3\sigma^2 - 7$

$3\sigma^2$

$9\sigma^2$

$9\sigma^2 - 7$

Correction:

9\sigma^2

$\mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X)$ . Constants shift the mean but do not affect spread, and scaling by $a$ scales the variance by $a^2$ . Hence $\mathrm{Var}(3X - 7) = 9\sigma^2$ .

Random variable

A measurable function $X : \Omega \to \mathbb{R}$ mapping outcomes of a random experiment to real numbers. Enables computation of probabilities, expectations, and distributions.

Probability density function (PDF)

A non-negative function $f_X(x)$ whose integral over any interval $[a, b]$ gives $P(a < X \le b)$ . It is the derivative of the CDF wherever the derivative exists.

Cumulative distribution function (CDF)

The function $F_X(x) = P(X \le x)$ , which completely characterises the distribution of a random variable. It is non-decreasing, right-continuous, with limits $0$ and $1$ .

Expectation

The weighted average $E[X] = \int x\,f_X(x)\,dx$ (continuous) or $E[X] = \sum_x x\,p_X(x)$ (discrete). A linear functional of the random variable's distribution.

Variance

$\mathrm{Var}(X) = E[(X - E[X])^2]$ , a measure of the spread of $X$ around its mean. Equals $E[X^2] - (E[X])^2$ . Units are the square of the units of $X$ .

Q-function

The tail probability of the standard normal distribution: $Q(x) = P(Z > x) = \frac{1}{2}\,\mathrm{erfc}(x/\sqrt{2})$ where $Z \sim \mathcal{N}(0,1)$ . Ubiquitous in error-probability expressions for digital modulation over AWGN channels.

Key Takeaway

The core message of this section in three bullets:

Random variables turn experiments into numbers. The measurable function $X : \Omega \to \mathbb{R}$ is the formal link between abstract probability spaces and the integrals, expectations, and variances that drive engineering design.
Six distributions cover most of wireless. Bernoulli and binomial for error counting, Poisson for arrivals, uniform for unknown phase, exponential for inter-arrival times and Rayleigh SNR, and Gaussian for thermal noise --- these six appear on almost every page of a communications textbook.
The Gaussian is the workhorse noise model. Because thermal noise is the sum of myriad independent microscopic contributions, the Central Limit Theorem guarantees it is Gaussian. The Q-function, the tail integral of the standard normal, is the single most important special function in digital communications: every BER formula you will encounter in Chapters 5--8 is built from it.

Common Mistake: Forgetting the $\frac{1}{2}$ in the Q-function vs erfc Relationship

Mistake:

Writing $Q(x) = \mathrm{erfc}(x/\sqrt{2})$ without the factor of $1/2$ , or equivalently $\mathrm{erfc}(x) = Q(x\sqrt{2})$ instead of $\mathrm{erfc}(x) = 2\,Q(x\sqrt{2})$ .

This error is extremely common in homework and even published papers. It leads to BER predictions that are off by exactly a factor of $2$ , which can silently propagate through simulation comparisons.

Correction:

The correct relations are:

$Q(x) = \frac{1}{2}\,\mathrm{erfc}\!\left(\frac{x}{\sqrt{2}}\right), \qquad \mathrm{erfc}(x) = 2\,Q\!\left(x\sqrt{2}\right).$

A useful sanity check: $Q(0) = 1/2$ and $\mathrm{erfc}(0) = 1$ . Without the factor of $1/2$ one would get $Q(0) = 1$ , which is clearly wrong since it would imply $P(Z > 0) = 1$ for a symmetric distribution.

Tip: Always verify your formula at $x = 0$ as a quick consistency test.

Common Mistake: Interpreting the PDF Value as a Probability

Mistake:

Claiming that $f_X(x_0)$ is the probability that $X = x_0$ . For continuous random variables, $P(X = x_0) = 0$ for every individual point. Yet it is common to see statements like "the probability of receiving $r = 1.5$ is $f_R(1.5) = 0.35$ ."

Correction:

The PDF is a density, not a probability. Only when integrated over an interval does it yield a probability:

$P(a < X \le b) = \int_a^b f_X(x)\,dx.$

Informally, $f_X(x_0)\,dx$ is the probability that $X$ falls in an infinitesimally small interval around $x_0$ . In particular, $f_X(x)$ can exceed $1$ (e.g., $\mathrm{Uniform}(0, 0.5)$ has $f_X(x) = 2$ on its support).

Random Variables and Distributions

Why Random Variables?

Definition: Random Variable

Definition: Cumulative Distribution Function (CDF)

Definition: Probability Density Function (PDF)

Definition: Probability Mass Function (PMF)

Definition: Expectation (Mean)

Definition: Variance and Standard Deviation

Theorem: Properties of Expectation

Linearity

Monotonicity

Product rule for independent RVs

Definition: Bernoulli Distribution

Definition: Binomial Distribution

Definition: Poisson Distribution

Definition: Continuous Uniform Distribution

Definition: Exponential Distribution

Definition: Gaussian (Normal) Distribution

Gaussian PDF and CDF Explorer

Parameters

Example: Bit-Error Rate of BPSK in AWGN

Condition on transmitted symbol

Standardise the Gaussian

Express in terms of the Q-function

Equivalent erfc form

The Gaussian Distribution: Shape, Scale, and the 68-95-99.7 Rule

Why This Matters: The Q-Function and Error Probability in Digital Communications

Quick Check

Quick Check

Quick Check

Random variable

Probability density function (PDF)

Cumulative distribution function (CDF)

Expectation

Variance

Q-function

Key Takeaway

Common Mistake: Forgetting the 12\frac{1}{2}21​ in the Q-function vs erfc Relationship

Common Mistake: Interpreting the PDF Value as a Probability

Definition:
Random Variable

Definition:
Cumulative Distribution Function (CDF)

Definition:
Probability Density Function (PDF)

Definition:
Probability Mass Function (PMF)

Definition:
Expectation (Mean)

Definition:
Variance and Standard Deviation

Definition:
Bernoulli Distribution

Definition:
Binomial Distribution

Definition:
Poisson Distribution

Definition:
Continuous Uniform Distribution

Definition:
Exponential Distribution

Definition:
Gaussian (Normal) Distribution

Common Mistake: Forgetting the $\frac{1}{2}$ in the Q-function vs erfc Relationship