Exploring Fundamental Probability Distributions: Bernoulli, Poisson, and Uniform

Understanding probability distributions forms the foundation of statistical modeling and data analysis. This tutorial examines three fundamental distributions through practical examples, showing how to identify the right distribution for different scenarios and calculate key probabilities.

When to Use Each Distribution

Bernoulli Distribution: Models single trials with two outcomes

• Website conversions (buy/don’t buy)
• Free throw attempts (make/miss)
• Quality control (pass/fail)

Poisson Distribution: Counts events over fixed intervals

• Customer service calls per hour
• Website page views per minute
• Defects per manufactured item

Uniform Distribution: All outcomes equally likely

• Die rolls
• Random number generation
• Lottery drawings

Bernoulli Distribution: Binary Outcomes

The Bernoulli distribution handles the simplest probability scenario: one trial, two outcomes.

Example: Website Conversion

A customer visits your e-commerce site with a 3% purchase probability.

Setup: X = 1 (purchase) with probability p = 0.03, X = 0 (no purchase) with probability 1 - p = 0.97

Key Formulas:

• Expected value: E(X) = p = 0.03
• Probability of no purchase: P(X = 0) = 1 - p = 0.97

The Bernoulli distribution’s elegance lies in its simplicity—the expected value equals the success probability.

Poisson Distribution: Counting Events

Use Poisson when counting random events occurring at a known average rate over fixed intervals.

Example: Call Center Operations

A call center receives 15 calls per hour. Find the probability of exactly 3 calls in 15 minutes.

Critical Step: Scale the rate for your time period

• Original rate: 15 calls/hour
• 15-minute rate: λ = 15 × (15/60) = 3.75 calls

Formula: P(X = k) = (λ^k × e^(-λ)) / k!

Calculation: P(X = 3) = (3.75³ × e^(-3.75)) / 3!

• 3.75³ ≈ 52.7
• e^(-3.75) ≈ 0.0235
• 3! = 6
• Result: ≈ 0.2066 or 20.66%

Uniform Distribution: Equal Probabilities

When all outcomes have equal likelihood, use the uniform distribution.

Example: Standard Die Roll

Rolling a six-sided die once.

Setup: Each outcome (1, 2, 3, 4, 5, 6) has probability 1/6

Multiple Outcomes: For rolling 2, 3, or 4: P(2 or 3 or 4) = P(2) + P(3) + P(4) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2

Since these events are mutually exclusive (you can’t roll multiple numbers simultaneously), add their individual probabilities.

Practical Application Tips

Distribution Selection

1. Count outcomes: Two outcomes suggest Bernoulli, multiple equal outcomes suggest uniform
2. Identify the interval: Fixed time/space intervals with event counting indicate Poisson
3. Check assumptions: Ensure independence between trials and constant probabilities

Common Calculations

• Bernoulli: Expected value = success probability
• Poisson: Always scale rates to match your time interval
• Uniform: Add probabilities for multiple favorable outcomes

Avoiding Mistakes

• Rate scaling: Convert Poisson rates to match your analysis period
• Mutually exclusive events: Only add probabilities when events cannot occur simultaneously
• Parameter identification: Clearly define success/failure before applying formulas

Next Steps

Master these three distributions before advancing to more complex ones like binomial (multiple Bernoulli trials) or exponential (time between Poisson events). Practice identifying distribution types from problem descriptions—this skill proves essential for real-world statistical modeling.

Start with simple examples, verify your calculations, and gradually tackle more complex scenarios involving multiple parameters or combined distributions.