Random Variables and Distributions - Examples and Constructions

Understanding standard distributions through examples provides intuition for modeling real-world phenomena. Each distribution has characteristic features suited to particular applications.

Bernoulli Distribution

The simplest random variable represents a single trial with two outcomes.

Definition

$X \sim \text{Bernoulli}(p)$ takes values in $\{0,1\}$ with: $P(X = 1) = p, \quad P(X = 0) = 1-p$

PMF: $p_X(k) = p^k(1-p)^{1-k}$ for $k \in \{0,1\}$

Models: coin flip, success/failure, yes/no questions.

Binomial Distribution

Definition

$X \sim \text{Binomial}(n,p)$ represents the number of successes in $n$ independent Bernoulli $(p)$ trials: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n$

Example

Toss a fair coin 10 times. Let $X$ = number of heads. Then $X \sim \text{Binomial}(10, 0.5)$ .

Probability of exactly 6 heads: $P(X = 6) = \binom{10}{6} (0.5)^{10} = 210 \times \frac{1}{1024} \approx 0.205$

Geometric Distribution

Definition

$X \sim \text{Geometric}(p)$ represents the number of trials until the first success: $P(X = k) = (1-p)^{k-1} p, \quad k = 1, 2, 3, \ldots$

Memoryless Property: $P(X > n+m | X > n) = P(X > m)$

The geometric distribution is the only discrete distribution with this property.

Poisson Distribution

Definition

$X \sim \text{Poisson}(\lambda)$ models the number of events in a fixed interval when events occur at constant rate $\lambda$ : $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0, 1, 2, \ldots$

Example

A call center receives an average of 5 calls per hour. The number of calls $X$ in an hour follows $\text{Poisson}(5)$ .

Probability of exactly 3 calls: $P(X = 3) = \frac{e^{-5} \cdot 5^3}{3!} = \frac{e^{-5} \cdot 125}{6} \approx 0.1404$

The Poisson distribution approximates Binomial $(n,p)$ when $n$ is large, $p$ is small, and $np = \lambda$ .

Uniform Distribution

Definition

$X \sim \text{Uniform}(a,b)$ is equally likely to take any value in $[a,b]$ : $f_X(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$

CDF: $F_X(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$

Exponential Distribution

Definition

$X \sim \text{Exponential}(\lambda)$ models waiting times between events in a Poisson process: $f_X(x) = \lambda e^{-\lambda x}, \quad x \geq 0$

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

Memoryless Property: $P(X > s+t | X > s) = P(X > t)$

The exponential distribution is the only continuous distribution with this property.

Example

The time until the next customer arrives follows $\text{Exponential}(2)$ (average 0.5 minutes).

Probability of waiting more than 1 minute: $P(X > 1) = e^{-2 \cdot 1} = e^{-2} \approx 0.1353$

Normal (Gaussian) Distribution

Definition

$X \sim \mathcal{N}(\mu, \sigma^2)$ has PDF: $f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

The standard normal $Z \sim \mathcal{N}(0,1)$ has $\mu = 0, \sigma = 1$ .

The normal distribution is ubiquitous due to the Central Limit Theorem. It approximates the sum of many independent random variables.

Remark

Each distribution family addresses specific modeling needs. Discrete distributions (Bernoulli, Binomial, Geometric, Poisson) count occurrences, while continuous distributions (Uniform, Exponential, Normal) measure continuous quantities. Choosing the right distribution requires understanding the underlying data-generating process.