Electronics Guide

Probability Distributions in Reliability

Probability distributions form the mathematical foundation for describing and predicting the failure behavior of electronic components and systems. These statistical models characterize how failure times are distributed across a population, enabling engineers to estimate reliability metrics, plan testing programs, and make informed design decisions. Selecting the appropriate distribution for a given application is essential for accurate reliability analysis.

Different failure mechanisms produce different patterns of failure times, and each probability distribution captures specific patterns. The exponential distribution describes random failures with constant failure rate. The Weibull distribution flexibly models infant mortality, random failure, and wear-out behavior. The lognormal distribution suits degradation and fatigue failures. Understanding these distributions and their applications equips reliability engineers to analyze data effectively and draw valid conclusions about product reliability.

Fundamental Reliability Functions

Before examining specific distributions, understanding the fundamental functions used to characterize reliability provides necessary foundation.

Probability Density Function

The PDF describes the distribution of failure times:

  • Definition: f(t) represents the probability density of failure at time t
  • Interpretation: f(t)dt gives probability of failure in small interval (t, t+dt)
  • Normalization: Integral of f(t) from 0 to infinity equals 1
  • Non-negative: f(t) greater than or equal to 0 for all t
  • Shape: Shape of PDF reveals failure time distribution pattern

The PDF shows where failures are concentrated in time, revealing underlying failure behavior.

Cumulative Distribution Function

The CDF gives cumulative failure probability:

  • Definition: F(t) = P(T less than or equal to t), probability of failure by time t
  • Relationship: F(t) equals integral of f(x) from 0 to t
  • Boundaries: F(0) = 0 and F(infinity) = 1
  • Monotonicity: F(t) is non-decreasing; failures accumulate over time
  • Unreliability: F(t) is also called the unreliability function

The CDF directly answers the question of what fraction of a population will have failed by a given time.

Reliability Function

The reliability function gives survival probability:

  • Definition: R(t) = P(T > t), probability of surviving beyond time t
  • Relationship: R(t) = 1 - F(t)
  • Boundaries: R(0) = 1 (all units start operational) and R(infinity) = 0 (all eventually fail)
  • Decreasing: R(t) decreases as time increases; reliability declines with age
  • Key metric: R(t) is often the primary reliability specification

The reliability function provides the probability that a unit will still be operational at any given time.

Hazard Function

The hazard function describes instantaneous failure rate:

  • Definition: h(t) = f(t)/R(t), conditional failure rate at time t given survival to t
  • Interpretation: h(t)dt gives probability of failure in next dt given survival to t
  • Units: Expressed in failures per unit time (e.g., failures per hour)
  • Shape significance: Increasing, constant, or decreasing hazard indicates wear-out, random, or infant mortality
  • Cumulative hazard: H(t) = integral of h(x) from 0 to t; R(t) = exp(-H(t))

The hazard function reveals how failure risk changes with age, indicating underlying failure mechanisms.

The Exponential Distribution

The exponential distribution is the simplest and most widely used distribution in reliability engineering, characterized by a constant failure rate.

Distribution Functions

Key functions for the exponential distribution:

  • PDF: f(t) = lambda * exp(-lambda*t) for t >= 0
  • CDF: F(t) = 1 - exp(-lambda*t)
  • Reliability: R(t) = exp(-lambda*t)
  • Hazard: h(t) = lambda (constant)
  • Parameter: lambda is the failure rate (failures per unit time)

The single parameter lambda completely characterizes the exponential distribution.

Key Properties

The exponential distribution has distinctive properties:

  • Constant failure rate: The probability of failure in the next time interval is independent of age
  • Memoryless property: P(T > t+s | T > t) = P(T > s); past survival does not affect future
  • Mean: MTTF = 1/lambda
  • Standard deviation: Also equals 1/lambda; coefficient of variation equals 1
  • Median: t_50 = ln(2)/lambda approximately equals 0.693/lambda

The memoryless property means equipment is as good as new at any age, which may or may not reflect reality.

Applications and Limitations

When to use and when to avoid the exponential distribution:

  • Random failures: Appropriate when failures result from random external events or overstress
  • Complex systems: Superposition of many failure mechanisms may produce approximately constant rate
  • Useful life region: Valid during the constant failure rate portion of bathtub curve
  • Not for wear-out: Inappropriate when failure rate increases with age
  • Not for infant mortality: Inappropriate when failure rate decreases with age

The exponential distribution is often assumed for convenience but should be validated against actual data.

Reliability Calculations

Common calculations using the exponential distribution:

  • Reliability at time t: R(t) = exp(-lambda*t) = exp(-t/MTTF)
  • Failure probability: F(t) = 1 - exp(-lambda*t)
  • Time for given reliability: t = -ln(R)/lambda
  • MTTF from test: MTTF = total operating time / number of failures
  • Confidence bounds: Chi-square based bounds for failure rate and MTTF

The mathematical simplicity of exponential calculations makes it attractive for preliminary analysis.

The Weibull Distribution

The Weibull distribution is the most versatile distribution in reliability engineering, capable of modeling decreasing, constant, and increasing failure rates.

Distribution Functions

Key functions for the two-parameter Weibull distribution:

  • PDF: f(t) = (beta/eta) * (t/eta)^(beta-1) * exp(-(t/eta)^beta)
  • CDF: F(t) = 1 - exp(-(t/eta)^beta)
  • Reliability: R(t) = exp(-(t/eta)^beta)
  • Hazard: h(t) = (beta/eta) * (t/eta)^(beta-1)
  • Parameters: beta (shape) and eta (scale, characteristic life)

The two parameters provide flexibility to fit diverse failure patterns.

Shape Parameter Interpretation

The beta parameter determines failure rate behavior:

  • Beta less than 1: Decreasing failure rate; indicates infant mortality
  • Beta equals 1: Constant failure rate; reduces to exponential distribution
  • Beta greater than 1: Increasing failure rate; indicates wear-out
  • Beta equals 2: Linear increasing hazard; Rayleigh distribution
  • Beta equals 3.4: Approximates normal distribution shape

The shape parameter directly indicates whether the failure mode is infant mortality, random, or wear-out.

Scale Parameter and Characteristic Life

The eta parameter determines when failures occur:

  • Definition: Eta is the time at which 63.2% of population has failed
  • Reliability at eta: R(eta) = exp(-1) approximately equals 0.368
  • Scale effect: Changing eta stretches or compresses the time axis
  • Mean relationship: Mean = eta * Gamma(1 + 1/beta)
  • B10 life: B10 = eta * (-ln(0.90))^(1/beta)

The characteristic life provides a reference point for comparing reliability across products.

Three-Parameter Weibull

Adding a location parameter extends flexibility:

  • Third parameter: Gamma represents minimum life before failures can occur
  • Modified functions: Replace t with (t-gamma) in all formulas
  • Threshold effect: No failures occur before time gamma
  • Physical interpretation: Represents guaranteed minimum life
  • Estimation challenge: Three parameters are harder to estimate reliably

Use the three-parameter Weibull when physical reasoning suggests a failure-free period.

Weibull Plotting

Graphical analysis reveals distribution fit and parameters:

  • Weibull paper: Log-log plot linearizes Weibull CDF
  • Y-axis: ln(ln(1/(1-F(t)))) or equivalent Weibull scale
  • X-axis: ln(t) or logarithmic time scale
  • Slope: Slope of fitted line equals beta
  • Intercept: Eta read from time axis at F = 63.2%

Weibull plotting provides visual assessment of fit and quick parameter estimation.

The Lognormal Distribution

The lognormal distribution models failures resulting from multiplicative degradation processes and is particularly useful for semiconductor and fatigue applications.

Distribution Functions

Key functions for the lognormal distribution:

  • PDF: f(t) = (1/(t*sigma*sqrt(2*pi))) * exp(-(ln(t)-mu)^2/(2*sigma^2))
  • CDF: F(t) = Phi((ln(t)-mu)/sigma), where Phi is standard normal CDF
  • Reliability: R(t) = 1 - Phi((ln(t)-mu)/sigma)
  • Hazard: h(t) = f(t)/R(t); initially increases then decreases
  • Parameters: mu (log-mean) and sigma (log-standard deviation)

The parameters describe the distribution of ln(t), which is normally distributed.

Key Properties

Distinctive characteristics of the lognormal distribution:

  • Right skewed: Long tail toward longer times; mean exceeds median
  • Median: t_50 = exp(mu); the median equals the geometric mean
  • Mean: Mean = exp(mu + sigma^2/2)
  • Mode: Mode = exp(mu - sigma^2)
  • Hazard shape: Hazard function increases then decreases; not monotonic

The non-monotonic hazard distinguishes the lognormal from Weibull for wear-out applications.

Physical Basis

The lognormal arises from multiplicative processes:

  • Degradation processes: Cumulative damage from many small multiplicative effects
  • Fatigue: Crack growth governed by multiplicative increments
  • Electromigration: Time to failure follows lognormal in many studies
  • Dielectric breakdown: Oxide breakdown often fits lognormal distribution
  • Central limit theorem: Product of many random variables tends toward lognormal

When failure results from accumulated multiplicative effects, lognormal is theoretically justified.

Comparison with Weibull

Understanding when to use lognormal versus Weibull:

  • Hazard behavior: Weibull hazard is monotonic; lognormal hazard peaks and decreases
  • Tail behavior: Lognormal has heavier right tail than most Weibull shapes
  • Parameter interpretation: Weibull parameters more directly interpretable
  • Extrapolation: Both can give very different predictions when extrapolating
  • Mechanism basis: Choose based on physical failure mechanism when possible

For short extrapolations, Weibull and lognormal often fit data equally well; mechanism knowledge guides selection.

The Normal Distribution

The normal (Gaussian) distribution has limited direct application in reliability but appears in certain contexts and underpins many statistical methods.

Distribution Functions

Key functions for the normal distribution:

  • PDF: f(t) = (1/(sigma*sqrt(2*pi))) * exp(-(t-mu)^2/(2*sigma^2))
  • CDF: F(t) = Phi((t-mu)/sigma)
  • Reliability: R(t) = 1 - Phi((t-mu)/sigma)
  • Hazard: h(t) = f(t)/R(t); increases without bound
  • Parameters: mu (mean) and sigma (standard deviation)

The symmetric bell shape is familiar but not typical of reliability applications.

Limitations for Reliability

The normal distribution has significant limitations:

  • Negative values: Normal allows negative failure times, which are physically impossible
  • Symmetric: Failure time distributions are typically right-skewed, not symmetric
  • Unbounded hazard: Hazard increases without bound, which may not be realistic
  • Limited flexibility: Fixed shape provides no flexibility unlike Weibull
  • Approximation only: May approximate tight wear-out distributions but rarely best choice

Use normal distribution cautiously and only when justified by data or physical reasoning.

Valid Applications

Contexts where normal distribution applies:

  • Wear-out with tight spread: When wear-out occurs in narrow time range relative to mean
  • Parameter variation: Component parameter distributions often normal
  • Degradation amount: Amount of degradation at fixed time may be normal
  • Log-transformed data: If ln(t) is normal, t is lognormal
  • Statistical inference: Underpins confidence intervals and hypothesis tests

Normal distribution remains important for statistical methods even when failure times are not normal.

Other Distributions

Additional distributions address specific reliability applications beyond the primary four.

Gamma Distribution

The gamma distribution generalizes the exponential:

  • Parameters: Shape parameter k and rate parameter lambda (or scale theta = 1/lambda)
  • Special cases: Exponential when k=1; Erlang when k is integer
  • Sum of exponentials: Sum of k exponential times follows Erlang distribution
  • Hazard: Increasing, constant, or decreasing depending on k
  • Applications: Queuing models, system with stages of failure

The gamma distribution is mathematically related to the Weibull but has different properties.

Inverse Gaussian Distribution

Models first passage time to a threshold:

  • Physical model: Time for degradation to reach failure threshold with drift and diffusion
  • Parameters: Mean mu and shape lambda
  • Hazard: Initially increases rapidly then approaches constant
  • Applications: Degradation modeling, wear processes
  • Relationship: Related to Wiener process with drift

Inverse Gaussian has theoretical basis for degradation-to-threshold failure models.

Birnbaum-Saunders Distribution

Another degradation-based distribution:

  • Origin: Derived from fatigue crack growth under cyclic loading
  • Parameters: Shape alpha and scale beta
  • Relationship: Related to normal distribution through transformation
  • Hazard: Initially increasing, eventually approaches constant
  • Applications: Material fatigue, crack propagation

Birnbaum-Saunders provides physically-motivated alternative to lognormal for fatigue applications.

Mixed Distributions

Mixtures model populations with multiple failure modes:

  • Concept: Overall distribution is weighted combination of component distributions
  • Defective subpopulation: Mix of early-failure defectives and normal units
  • Multiple mechanisms: Different failure modes with different distributions
  • Parameter estimation: More complex; may require EM algorithm or numerical methods
  • Identification: Curvature on Weibull plots may indicate mixed population

Mixed distributions capture real-world situations where multiple distinct failure populations exist.

Distribution Selection and Validation

Choosing the right distribution and validating the choice is critical for accurate reliability analysis.

Selection Criteria

Factors guiding distribution selection:

  • Physical mechanism: Select distribution consistent with known failure physics
  • Failure rate pattern: Choose distribution matching observed hazard behavior
  • Data fit: Select distribution that best fits available failure data
  • Extrapolation needs: Consider extrapolation behavior if predicting beyond data range
  • Precedent: Use distributions proven for similar applications

Physical understanding should guide selection when possible; statistical fit alone can be misleading.

Goodness of Fit Tests

Statistical tests evaluate distribution fit:

  • Chi-square test: Compare observed and expected frequencies in intervals
  • Kolmogorov-Smirnov test: Maximum difference between empirical and theoretical CDFs
  • Anderson-Darling test: Weighted K-S test emphasizing tail fit
  • Correlation coefficient: Correlation on probability plot indicates fit quality
  • Likelihood ratio: Compare likelihoods of competing distributions

Formal tests quantify fit quality but should be complemented by graphical assessment.

Graphical Methods

Visual assessment reveals fit characteristics:

  • Probability plots: Data should follow straight line on appropriate probability paper
  • Hazard plots: Plot estimated hazard against theoretical hazard function
  • Q-Q plots: Quantile-quantile plots compare data and theoretical quantiles
  • Residual plots: Examine residuals for systematic patterns
  • Multiple distributions: Compare same data on different probability papers

Graphical methods reveal patterns and outliers that numerical tests may miss.

Sensitivity Analysis

Evaluate impact of distribution choice:

  • Compare predictions: Calculate reliability metrics with different distributions
  • Extrapolation sensitivity: Assess how far predictions diverge beyond data range
  • Parameter uncertainty: Propagate parameter uncertainty through predictions
  • Consequence assessment: Determine if distribution choice matters for decisions
  • Conservative choice: When uncertain, choose distribution giving conservative predictions

If different reasonable distributions give similar predictions, distribution choice may not be critical.

Summary

Probability distributions provide the mathematical framework for describing how failure times are distributed across product populations. The exponential distribution with its constant failure rate offers simplicity and is appropriate for random failures during useful life. The Weibull distribution provides flexibility to model infant mortality, random failure, and wear-out behavior through its shape parameter. The lognormal distribution suits degradation and fatigue mechanisms arising from multiplicative processes. The normal distribution has limited direct reliability application but underpins many statistical methods.

Each distribution has characteristic properties that make it suitable for specific applications. Understanding these properties enables proper distribution selection based on physical failure mechanisms, data characteristics, and analysis requirements. The hazard function is particularly informative: constant hazard suggests exponential, decreasing hazard indicates infant mortality with Weibull beta less than one, and increasing hazard indicates wear-out with Weibull beta greater than one or potentially lognormal.

Distribution selection should be guided by physical understanding when possible, supported by statistical fit assessment. Graphical methods complement formal goodness-of-fit tests by revealing patterns and anomalies. Sensitivity analysis determines whether distribution choice significantly impacts conclusions. When properly selected and validated, probability distributions enable accurate reliability prediction, informed design decisions, and effective reliability demonstration testing.