Common Random Variables - Geophysical Data Analysis Book

Learning Objectives¶

Understand the definitions and basic properties of the most common random variables used in probability, statistics, and geophysics.
Work with CDFs/PDFs and expectations for:
- Bernoulli
- Binomial
- Geometric
- Uniform
- Exponential
- Gaussian (Normal)
Connect these distributions to modeling choices in geophysical problems.

Motivation¶

Many models in geophysics involve simple stochastic components:

Sensor failures → Bernoulli
Number of detections in repeated trials → Binomial
Number of seismic events until the first detection → Geometric
Modeling unknown phase or angle → Uniform
Waiting times between nuclear decay events → Exponential
Measurement noise → Gaussian

These “canonical” distributions form the backbone of probabilistic modeling.
We introduce each with:

Definition
PDF
Expectation & variance
Interpretation & use cases

Discrete Random Variables¶

Bernoulli Random Variable¶

A Bernoulli random variable models a single binary outcome (success/failure).

Definition¶

Let

X \sim \text{Bernoulli}(p), \quad p \in [0,1].

Then

\mathbb{P}(X=1)=p, \qquad \mathbb{P}(X=0)=1-p.

Expectation and Variance¶

\mathbb{E}[X]=p, \qquad \mathrm{Var}(X)=p(1-p).

Interpretation¶

Models a “yes/no” event: arrival/no arrival, failure/success, detection/no detection.
Often used as building blocks for more complex models (e.g., Binomial, Bernoulli processes).

Geometric Random Variable¶

The geometric distribution measures the number of independent Bernoulli trials needed until the first success.

Definition¶

Let

X \sim \text{Geom}(p).

Then

\mathbb{P}(X=k) = (1-p)^{k-1}p.

Expectation and Variance¶

\mathbb{E}[X] = \frac{1}{p}, \qquad \mathrm{Var}(X) = \frac{1-p}{p^2}.

Interpretation¶

Models waiting time (in counts) until a success.
Example: number of seismic traces until the first clear reflection.

Binomial Random Variable¶

The binomial distribution counts the number of successes in a fixed number of independent Bernoulli trials.

Definition¶

Let

X \sim \text{Binomial}(n,p), \quad n \in \mathbb{N}, \; p \in [0,1].

Then for $k=0,1,\dots,n$ ,

\mathbb{P}(X=k) = \binom{n}{k} p^k (1-p)^{n-k}.

Expectation and Variance¶

\mathbb{E}[X] = np, \qquad \mathrm{Var}(X) = np(1-p).

Interpretation¶

Models the count of successes in a fixed number of repeated trials.
Example: number of stations that successfully detect an event out of $n$ stations, assuming each station detects independently with probability $p$ .
Can be viewed as the sum of $n$ independent Bernoulli random variables.

Continuous Random Variables¶

Uniform Distribution¶

The simplest continuous distribution: all values in an interval are equally likely.

Definition¶

Let

X \sim \text{Uniform}(a,b),\quad a < b.

PDF:

f_X(x)= \begin{cases} \frac{1}{b-a}, & a \le x \le b, \\ 0, & \text{otherwise}. \end{cases}

Expectation and Variance¶

\mathbb{E}[X]=\frac{a+b}{2}, \qquad \mathrm{Var}(X)=\frac{(b-a)^2}{12}.

Interpretation¶

Unknown phase, unknown time shift, random orientation.
Often serves as a least-informative prior over bounded ranges.

Exponential Distribution¶

The exponential models waiting times between independent events (e.g., Poisson processes).

Definition¶

Let

X \sim \text{Exponential}(\lambda), \quad \lambda > 0.

PDF:

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

Expectation and Variance¶

\mathbb{E}[X]=\frac{1}{\lambda}, \qquad \mathrm{Var}(X)=\frac{1}{\lambda^2}.

Memoryless Property¶

\mathbb{P}(X > s+t \mid X > s) = \mathbb{P}(X > t).

Proof (click to expand)

For $X\sim \mathrm{Exponential}(\lambda)$ and $s,t\ge 0$ :

\mathbb{P}(X>u)=e^{-\lambda u}, \quad u\ge 0.

So,

\begin{align} &\mathbb{P}(X>s+t\mid X>s)\\ =&\frac{\mathbb{P}(X>s+t, X>s)}{\mathbb{P}(X>s)}\\ =&\frac{\mathbb{P}(X>s+t)}{\mathbb{P}(X>s)}\\ =&\frac{e^{-\lambda(s+t)}}{e^{-\lambda s}}\\ =&e^{-\lambda t}\\ =&\mathbb{P}(X>t). \end{align}

Hence the exponential distribution is memoryless.

Interpretation¶

Time between independent seismic events.
Waiting time until a Poisson-distributed arrival.
Common in queueing, signal processing, and reliability modeling.

Gaussian (Normal) Distribution¶

The most important continuous distribution in modeling measurement noise and aggregated effects.

Definition¶

Let

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

PDF:

f_X(x)= \frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).

Expectation and Variance¶

\mathbb{E}[X] = \mu, \qquad \mathrm{Var}(X) = \sigma^2.

Interpretation¶

Models additive noise in seismic recordings.
Arises from many small independent perturbations.
Stable under linear transformations and sums.

Key Points¶

Bernoulli models a single success/failure outcome.
Binomial models the number of successes in a fixed number of trials.
Geometric models waiting time until the first success.
Uniform models complete uncertainty on a bounded interval.
Exponential models memoryless waiting times.
Gaussian models aggregated random effects and noise.