Ferkans — Interactive Telecom Tutor

When a Distribution Is Neither Discrete Nor Continuous

Many real-world random variables do not fit neatly into the discrete or continuous categories. A wireless channel may be completely blocked (zero SNR, a discrete outcome with positive probability) or have a continuously distributed SNR when active. The CDF handles all cases uniformly — it is the true universal descriptor — and the Dirac delta function provides a notational device to write a unified "density" even for mixed distributions.

Definition:
Mixed Distribution

A random variable $X$ has a mixed distribution if its CDF $F$ is piecewise continuous with a countable number of jumps. There exist a continuous CDF $F_0$ , a discrete CDF $F_1$ , and a parameter $q \in (0, 1)$ such that

$F(x) = q\,F_0(x) + (1-q)\,F_1(x).$

The jump at each point $x_k$ has size $\mathbb{P}(X = x_k) > 0$ (the discrete component), while the continuous part contributes probability through a density.

The decomposition reflects the law of total probability: with probability $q$ , $X$ is drawn from the continuous component, and with probability $1-q$ , from the discrete component.

Definition:
Dirac Delta in Density Expressions

The Dirac delta function $\delta(x)$ is a generalized function satisfying:

$\delta(x) = 0$ for $x \neq 0$ .
$\int_{-\infty}^{\infty} \delta(x)\,dx = 1$ .
For any continuous function $h$ : $\int_{-\infty}^{\infty} h(x)\,\delta(x - a)\,dx = h(a)$ .

Using the Dirac delta, a mixed distribution with continuous density $f_0$ and point masses $p_k$ at locations $x_k$ can be written as a single "generalized density":

$f(x) = q\,f_0(x) + (1-q)\sum_k p_k\,\delta(x - x_k).$

The Dirac delta is not a function in the classical sense but a distribution (in the sense of Schwartz). Its use in probability is a convenient notation that allows us to treat all distributions — discrete, continuous, and mixed — with a single integral formalism.

Example: The Blocked Channel

A wireless channel is blocked (zero gain) with probability $p$ and has Rayleigh-distributed gain $R$ (with parameter $\sigma$ ) otherwise. Find the CDF and generalized PDF of the channel gain $G$ .

Solution

Model the random variable

$G = 0$ with probability $p$ , and $G = R \sim \text{Rayleigh}(\sigma)$ with probability $1 - p$ .

CDF

For $g < 0$ : $F_G(g) = 0$ . For $g = 0$ : $F_G(0) = p$ (jump of size $p$ ). For $g > 0$ : $F_G(g) = p + (1-p)\left(1 - e^{-g^2/(2\sigma^2)}\right) = 1 - (1-p)e^{-g^2/(2\sigma^2)}$ .

Generalized PDF

$f_G(g) = p\,\delta(g) + (1-p)\frac{g}{\sigma^2}e^{-g^2/(2\sigma^2)}\,\mathbf{1}_{\{g > 0\}}$ .

The first term (Dirac delta at zero) captures the blocking event; the second term is the Rayleigh density weighted by the probability of being active.

Example: Clipped Gaussian (Rectifier)

Let $X \sim \mathcal{N}(0, 1)$ and define $Y = \max(X, 0)$ (a half-wave rectifier). Find the CDF and generalized PDF of $Y$ .

Solution

CDF of $Y$

For $y < 0$ : $F_Y(y) = 0$ . For $y = 0$ : $F_Y(0) = \mathbb{P}(Y \leq 0) = \mathbb{P}(X \leq 0) = 1/2$ . For $y > 0$ : $F_Y(y) = \mathbb{P}(\max(X,0) \leq y) = \mathbb{P}(X \leq y) = \Phi(y)$ .

Observe the jump

$F_Y$ has a jump of size $1/2$ at $y = 0$ (the event $\{X \leq 0\}$ , which maps to $Y = 0$ ). For $y > 0$ , $F_Y$ is continuous with derivative $\phi(y)$ .

Generalized PDF

$f_Y(y) = \frac{1}{2}\delta(y) + \phi(y)\,\mathbf{1}_{\{y > 0\}}$ , where $\phi(y) = \frac{1}{\sqrt{2\pi}}e^{-y^2/2}$ .

Mixed Distribution CDF Visualization

Visualize the CDF of a mixed distribution: a channel that is blocked with probability $p$ and Rayleigh-distributed otherwise. Observe the jump at $g = 0$ and the smooth rise for $g > 0$ .

Parameters

p

(blocking probability)0.3

\sigma

(Rayleigh parameter)1

Theorem: The CDF as the Universal Descriptor

For any random variable $X$ (discrete, continuous, mixed, or even singular), the CDF $F(x) = \mathbb{P}(X \leq x)$ is well-defined and satisfies:

$F$ is non-decreasing and right-continuous.
$\lim_{x \to -\infty}F(x) = 0$ , $\lim_{x \to \infty}F(x) = 1$ .
All probabilities can be computed from $F$ $F$ :
- $\mathbb{P}(a < X \leq b) = F(b) - F(a)$
- $\mathbb{P}(X = a) = F(a) - F(a^-)$
- $\mathbb{P}(X > a) = 1 - F(a)$

The CDF uniquely determines the distribution of $X$ .

Proof

Uniqueness

Two probability measures on $(\mathbb{R}, \mathcal{B})$ that agree on all half-lines $(-\infty, x]$ must agree on all Borel sets, by the $\pi$ - $\lambda$ theorem. Since $F$ determines $\mathbb{P}((-\infty, x])$ for all $x$ , it uniquely determines the entire probability measure. $\blacksquare$

Common Mistake: Computing Expectations for Mixed Distributions

Mistake:

Integrating only the continuous part of the density when computing $\mathbb{E}[X]$ for a mixed distribution, forgetting the discrete component.

Correction:

For a mixed RV $X$ with continuous density $f_0$ (weight $q$ ) and point masses $p_k$ at $x_k$ (weight $1-q$ ):

$\mathbb{E}[X] = q\int x\,f_0(x)\,dx + (1-q)\sum_k x_k\,p_k$ .

Both parts must be included. Equivalently, integrate $x$ against the generalized density $f(x) = q f_0(x) + (1-q)\sum_k p_k\,\delta(x - x_k)$ .

Quick Check

A mixed RV $X$ equals $0$ with probability $0.4$ and is $\text{Uniform}[0,2]$ otherwise. What is $F_X(1)$ ?

$0.4$

$0.7$

$0.5$

$0.9$

Correction:

0.7

$F_X(1) = 0.4 + 0.6 \cdot \frac{1}{2} = 0.4 + 0.3 = 0.7$ .

🔧Engineering Note

Simulating Mixed Random Variables

To generate samples from a mixed distribution: first draw $U \sim \text{Uniform}[0,1]$ . If $U < p$ (the discrete weight), output the discrete value. Otherwise, generate a sample from the continuous component using inverse transform sampling or any other method. This two-stage procedure directly implements the mixture decomposition and is the standard approach in Monte Carlo simulations of wireless channels with blocking.

Why This Matters: Mixed Distributions in Fading Channels

Mixed distributions arise naturally in wireless propagation. A channel may experience complete blockage (e.g., a vehicle entering a tunnel, or a building obstruction) with some probability, and continuous fading otherwise. The outage analysis of such channels requires the mixed CDF: $F_\gamma(\gamma_0) = p + (1-p)F_{\gamma|\text{active}}(\gamma_0)$ . The discrete jump at $\gamma = 0$ represents the irreducible outage floor that no amount of transmit power can overcome.

Dirac Delta Function

A generalized function $\delta(x)$ satisfying $\delta(x) = 0$ for $x \neq 0$ and $\int \delta(x)\,dx = 1$ . Used to represent point masses in the density formalism: $\mathbb{P}(X = a) = p$ is encoded as $p\,\delta(x - a)$ in the generalized PDF.

Key Takeaway

The CDF is the universal representation of a random variable's distribution. It handles discrete, continuous, and mixed distributions without case distinctions. The Dirac delta extends the density formalism to encompass point masses, allowing a single integral notation for all expectations and probabilities.

Historical Note: The Dirac Delta and the Mathematicians

1930–1950

Paul Dirac introduced his "delta function" in his 1930 textbook The Principles of Quantum Mechanics, treating it as an ordinary function with peculiar properties. Mathematicians initially objected — no classical function can be zero everywhere except at a point yet integrate to one. Laurent Schwartz's theory of distributions (1945–1950) provided the rigorous foundation, showing that the Dirac delta is a continuous linear functional on a space of test functions. The moral: physicists' and engineers' intuition about the delta function is correct, even if the formal justification requires some functional analysis.

Mixed Distributions and the CDF as the Unifying Object

When a Distribution Is Neither Discrete Nor Continuous

Definition: Mixed Distribution

Definition: Dirac Delta in Density Expressions

Example: The Blocked Channel

Model the random variable

CDF

Generalized PDF

Example: Clipped Gaussian (Rectifier)

CDF of $Y$

Observe the jump

Generalized PDF

Mixed Distribution CDF Visualization

Parameters

Theorem: The CDF as the Universal Descriptor

Uniqueness

Common Mistake: Computing Expectations for Mixed Distributions

Quick Check

Simulating Mixed Random Variables

Why This Matters: Mixed Distributions in Fading Channels

Dirac Delta Function

Key Takeaway

Historical Note: The Dirac Delta and the Mathematicians

Definition:
Mixed Distribution

Definition:
Dirac Delta in Density Expressions