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Common Probability Distributions

📊 Probability Theory 📅 Updated: Oct 24, 2025 ⏱️ 12 min read
In probability theory and statistics, a probability distribution describes the likelihood of possible outcomes for a random variable. Common distributions serve as foundational models for real-world phenomena, enabling hypothesis testing, predictive modeling, and statistical inference across scientific disciplines.

1. Normal (Gaussian) Distribution

The normal distribution is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is described by the probability density function (PDF):

f(x | μ, σ²) = (1 / (σ√(2π))) · exp(−(x − μ)² / (2σ²))
Parameters
μ (mean), σ² (variance)
Mean
μ
Variance
σ²
Support
−∞ < x < +∞
μ
Figure 1: Probability density curve of a standard normal distribution (μ=0, σ=1)

Applications: Heights, blood pressure, measurement errors, IQ scores, and any naturally occurring phenomenon influenced by many small independent factors (Central Limit Theorem).

2. Binomial Distribution

A discrete probability distribution describing the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success p.

P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
Parameters
n (trials), p (success prob)
Mean
n · p
Variance
n · p · (1 − p)
Support
k ∈ {0, 1, ..., n}
  • Quality control inspection (pass/fail testing)
  • Clinical trial success rates
  • Survey response modeling (yes/no questions)

3. Poisson Distribution

A discrete distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event.

P(X = k) = (λᵏ · e⁻λ) / k!
Parameter
λ (average rate)
Mean
λ
Variance
λ
Support
k ∈ {0, 1, 2, ...}
  • Call center arrival rates
  • Radiation decay counts
  • Network packet arrivals in fixed windows

4. Exponential Distribution

A continuous distribution that models the time between events in a Poisson process. It is memoryless, meaning the probability of an event occurring in the next time interval is independent of how much time has already elapsed.

f(x | λ) = λ · e⁻λx for x ≥ 0
Parameter
λ (rate)
Mean
1 / λ
Variance
1 / λ²
Support
x ≥ 0
  • Component failure times (reliability engineering)
  • Inter-arrival times in queueing systems
  • Radioactive decay intervals

5. Uniform Distribution

A family of probability distributions characterized by maximum entropy. In the continuous case, all intervals of the same length on the distribution's support are equally probable.

f(x | a, b) = 1 / (b − a) for a ≤ x ≤ b
Parameters
a (min), b (max)
Mean
(a + b) / 2
Variance
(b − a)² / 12
Support
a ≤ x ≤ b
  • Random number generation
  • Measurement rounding errors
  • Simulations requiring unbiased inputs

6. Summary Comparison

Distribution Type Parameters Mean Variance
Normal Continuous μ, σ² μ σ²
Binomial Discrete n, p np np(1−p)
Poisson Discrete λ λ λ
Exponential Continuous λ 1/λ 1/λ²
Uniform Continuous a, b (a+b)/2 (b−a)²/12

References & Further Reading

  1. Feller, W. (1968). An Introduction to Probability Theory and Its Applications (Vol. 1, 3rd ed.). Wiley.
  2. Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.
  3. Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
  4. NIST/SEMATECH e-Handbook of Statistical Methods. Retrieved from www.itl.nist.gov