Which distribution should I use?
If this question ever crossed your mind during problem-solving, you’re not alone. The trio Binomial, Poisson, and Normal distributions may seem confusing at first, but once you see their differences and connections clearly, solving probability problems becomes a breeze.
This comparison article aims to help students clearly distinguish between these three key distributions Binomial and Poisson (discrete), and Normal (continuous). It also explains when each distribution should be applied and how they relate to one another.
When to Use Each Distribution
- Binomial: When you're counting successes or failures across a fixed number of independent trials. (e.g., number of heads in 10 coin tosses)
- Poisson: When events are rare and occur independently over a continuous interval of time, space, or area. (e.g., number of emails received in an hour)
- Normal: When data are continuous and symmetrically distributed. Often used when sample sizes are large or when applying the Central Limit Theorem.
Shape, Mean & Variance
- Binomial: Symmetric if p = 0.5, skewed otherwise. Mean = np, Variance = np(1-p)
- Poisson: Skewed right (less skewed as mean increases). Mean = Variance = λ
- Normal: Perfectly symmetric bell curve. Mean = μ, Variance = σ²
Parameters
- Binomial: n (number of trials), p (probability of success)
- Poisson: λ (average rate of occurrence)
- Normal: μ (mean), σ² (variance)
Transition & Approximation
- Binomial → Poisson: When n is large and p is small, Binomial(n, p) ≈ Poisson(λ = np)
- Poisson → Normal: When λ is large (typically > 10), Poisson(λ) ≈ Normal(μ = λ, σ² = λ)
- Binomial → Normal: If n is large, and both np & n(1–p) ≥ 5, then Binomial(n, p) ≈ Normal(μ = np, σ² = np(1–p))
Comparison Table
| Feature | Binomial | Poisson | Normal |
|---|---|---|---|
| Type | Discrete | Discrete | Continuous |
| Parameters | n, p | λ | μ, σ² |
| Shape | Symmetric/Skewed | Right Skewed | Bell-shaped |
| Use Case | Fixed trials (e.g. surveys) | Rare events (e.g. arrivals) | Measurement data (e.g. scores) |
| Mean = Variance? | No | Yes | No |
Summary
- Binomial: Count of successes in fixed trials. Finite.
- Poisson: Count of events over time/space. Infinite but discrete.
- Normal: Measurement data with infinite possibilities and continuous range.
Final Call: Don’t just memorize ask yourself, “What am I counting or measuring?” That question alone will guide you to the right distribution more often than not.