Continuous Probability Distributions

Probability density functions, Normal, Uniform, Exponential, Gamma, Weibull, and Lognormal distributions.
Unlike discrete variables which are counted (e.g., number of cracks), continuous variables are measured (e.g., length, time, pressure, strength). Because a continuous variable can take on infinitely many values within a range, the probability of it taking any specific exact value is zero. Instead, we calculate the probability that the variable falls within a specified interval.

Probability Density Functions

The continuous analog to a probability mass function.

Probability Density Function (PDF), f(x)f(x)

A function describing the relative likelihood for a continuous random variable to take on a given value. The probability that XX lies between aa and bb is the area under the curve f(x)f(x) from aa to bb.
P(a≤X≤b)=∫abf(x) dx P(a \le X \le b) = \int_{a}^{b} f(x) \, dx
It must satisfy two conditions:
  • f(x)≥0f(x) \ge 0 for all xx.
  • ∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x) \, dx = 1 (the total area under the curve is 1).

The Normal Distribution

The most important continuous distribution in statistics.
The Normal (Gaussian) distribution is pervasive because many natural phenomena and measurement errors follow a bell-shaped curve.

Normal Distribution

A continuous, symmetric, bell-shaped distribution completely defined by its mean (μ\mu) and standard deviation (σ\sigma).
f(x)=1σ2πe−12(x−μσ)2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}
  • It is centered at μ\mu, which is also its median and mode.
  • The spread is determined by σ\sigma; a larger σ\sigma results in a flatter, wider curve.

The Standard Normal Distribution

Converting any normal distribution to a standard scale.
To calculate probabilities for any normal distribution, we standardize the variable XX into a ZZ-score, which represents the number of standard deviations xx is from the mean.

Z-Score

A dimensionless quantity used to standardly map any normal distribution to a Standard Normal Distribution (μ=0\mu = 0, σ=1\sigma = 1).
Z=X−μσ Z = \frac{X - \mu}{\sigma}
Once ZZ is found, probabilities P(Z≤z)P(Z \le z) are looked up in a Standard Normal Table (Z-table) or calculated via software.

Other Common Continuous Distributions

Models used for specific engineering scenarios, particularly in reliability and failure analysis.

The Uniform Distribution

Continuous Uniform Distribution

Used when all values within an interval [a,b][a, b] are equally likely. The PDF forms a rectangle. (e.g., Rounding errors in digital measurements).
f(x)=1b−afor a≤x≤b f(x) = \frac{1}{b - a} \quad \text{for } a \le x \le b
  • Mean: μ=a+b2\mu = \frac{a+b}{2}
  • Variance: σ2=(b−a)212\sigma^2 = \frac{(b-a)^2}{12}

The Exponential Distribution

Exponential Distribution

Closely related to the Poisson distribution. While Poisson models the number of occurrences in a fixed interval, the Exponential distribution models the time between occurrences (e.g., time between structural failures, or lifespan of a lightbulb).
f(x)=λe−λxfor x≥0,λ>0 f(x) = \lambda e^{-\lambda x} \quad \text{for } x \ge 0, \lambda > 0
  • Mean: μ=1λ\mu = \frac{1}{\lambda}
  • Variance: σ2=1λ2\sigma^2 = \frac{1}{\lambda^2}
Memoryless Property: The probability of failure in the next instant does not depend on how long the component has already survived.

The Gamma, Weibull, and Lognormal Distributions

Critical distributions for reliability engineering and material strength.

Gamma Distribution

A generalization of the Exponential distribution. It models the time until kk (the shape parameter, often denoted α\alpha) consecutive events occur, rather than just the first event.
f(x)=λαΓ(α)xα−1e−λxfor x>0 f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x} \quad \text{for } x > 0

Weibull Distribution

Extensively used in reliability engineering to model the "time to failure" of materials and mechanical systems (e.g., fatigue life of asphalt pavements or steel bearings). Unlike the memoryless Exponential distribution, Weibull can model failure rates that increase over time (wear-out) or decrease over time (early infant mortality).
f(x)=βη(xη)β−1e−(x/η)βfor x>0 f(x) = \frac{\beta}{\eta} \left( \frac{x}{\eta} \right)^{\beta-1} e^{-(x/\eta)^\beta} \quad \text{for } x > 0
  • β\beta (Shape parameter): If β>1\beta > 1, the failure rate increases over time (wear-out phase).
  • η\eta (Scale parameter or characteristic life): The time by which 63.2% of the population will have failed.

Lognormal Distribution

If a variable Y=ln⁡(X)Y = \ln(X) follows a Normal distribution, then XX follows a Lognormal distribution. It is widely used to model environmental data, such as stream flows, pollutant concentrations, and grain sizes in soils, because it is right-skewed and bounded at zero (variables cannot be negative).
Key Takeaways
  • PDFs: For continuous variables, probability is the area under the PDF curve. The probability of an exact value is zero.
  • Normal Distribution: The benchmark bell-shaped curve. Use ZZ-scores to standardize and find probabilities.
  • Uniform: Constant probability over an interval.
  • Exponential: Time between independent, random events. It is memoryless.
  • Gamma: Time until kk events occur.
  • Weibull: The standard for modeling material fatigue life and "time to failure" with changing failure rates.
  • Lognormal: Ideal for highly skewed positive data, like pollutant concentrations or streamflow.
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