6.13 Additional Distributions: Hypergeometric, Exponential, Lognormal, Weibull, and Bivariate

Choosing a distribution by situation

Hypergeometric

Models the number of successes in $n$ draws from a finite population of size $N$ containing $K$ successes, drawn without replacement.

$$P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$$

The distinguishing feature is that each draw changes the composition of the remainder, so the probability of success is not constant across draws. That is exactly the binomial assumption that fails.

When it matters. The rule of thumb is that the binomial is an adequate approximation when the sample is less than about 10% of the lot, $n / N < 0.1$. Below that the depletion effect is negligible; above it the binomial understates the true precision of the sample.

Example: a lot of 50 items contains 5 defectives and 10 are inspected without replacement. The probability the sample contains no defectives is hypergeometric, not binomial, because $10/50 = 0.2$ exceeds the threshold. This is the standard model behind acceptance sampling from finite lots.

Exponential

Models the time between events occurring at a constant average rate $\lambda$, and equivalently the life of an item with a constant failure rate.

$$f(t) = \lambda e^{-\lambda t}, \qquad R(t) = e^{-\lambda t}, \qquad \mu = \frac{1}{\lambda} = MTBF$$

The mean equals the standard deviation, and the distribution is strongly right-skewed.

The memoryless property is unique to the exponential among continuous distributions:

$$P(T > s + t \mid T > s) = P(T > t)$$

A component that has already survived 1,000 hours has exactly the same remaining life distribution as a new one. This is why preventive replacement is pointless for a truly exponential failure mode -- replacing a survivor with a new item does not improve the failure rate. That result is a favourite exam question.

The exponential pairs with the Poisson: if events occur as a Poisson process with rate $\lambda$, the waiting time between them is exponential with the same $\lambda$.

Lognormal

A variable $X$ is lognormal when $\ln(X)$ is normally distributed. It arises when an outcome is the product of many independent factors, just as the normal arises when an outcome is their sum.

Characteristics: strictly positive, right-skewed, with a long upper tail. It fits many Six Sigma variables well:

• Cycle times, repair times, and call handling times
• Time to failure for fatigue and crack-growth mechanisms
• Particle sizes, concentrations, and many laboratory assay results
• Income, claim amounts, and other economic quantities

Practical implication. Right-skewed data that fails a normality test is often lognormal, and taking natural logarithms frequently produces a normal distribution on which standard methods can be used. This is also the $\lambda = 0$ case of the Box-Cox transformation. For capability analysis on lognormal data, either transform first or use a distribution-fitting approach with percentile-based capability.

Weibull

The most flexible life-data distribution, with a shape parameter $\beta$ and a scale (characteristic life) parameter $\eta$:

$$R(t) = e^{-(t/\eta)^{\beta}}$$

Its power comes from $\beta$, which determines the shape of the hazard function -- the instantaneous failure rate.

$\beta \approx 3.4$ approximates a normal distribution, and $\beta = 2$ gives the Rayleigh distribution.

The characteristic life $\eta$ is the time by which 63.2% of the population has failed, regardless of $\beta$. Interpreting $\eta$ as the mean is a common error; the mean is $\eta \cdot \Gamma(1 + 1/\beta)$.

The reliability workflow is to fit a Weibull to the life data, read $\beta$ to identify the failure regime, and select the maintenance strategy accordingly. Scheduling preventive replacement on a component whose fitted $\beta$ is below 1 actively increases failures, because each replacement reintroduces an item into the infant-mortality region.

Bivariate normal

Describes two continuous variables that are jointly normal, characterized by $\mu_X$, $\mu_Y$, $\sigma_X$, $\sigma_Y$, and the correlation $\rho$. Its contours of constant density are ellipses, circular when $\rho = 0$ and increasingly elongated as $|\rho|$ approaches 1.

Two properties matter for Six Sigma:

• Joint normality is the assumption behind inference on the correlation coefficient and behind prediction intervals in simple linear regression.
• Under bivariate normality, zero correlation implies independence. This equivalence does not hold in general: two variables can have $\rho = 0$ and still be strongly related, for example when $Y = X^2$ over a symmetric range of $X$.

More broadly, the multivariate normal underlies the multivariate tools in the Analyze phase -- MANOVA, factor analysis, discriminant analysis, and Hotelling's $T^2$ multivariate control chart.

Quick recognition guide