Weibull Distribution

Compact study note.

Summary

The Weibull distribution is a flexible positive distribution often used for lifetimes and reliability. Its shape parameter controls increasing, constant, or decreasing hazard behavior.^[1]

Prerequisites

• Continuous Random Variable
• Gamma Function

Definition

$X\sim \mathrm{Weibull}(k,\lambda )$ with shape $k$ and scale $\lambda$ .

Notation and Assumptions

$k>0$ and $\lambda >0$ . This note uses $\lambda$ like scale, not rate.

Parameters

$k>0$ and $\lambda >0$ .

Support

$[0,\mathrm{\infty })$ .

PMF or PDF

$f_X(x)=(k/\lambda )(x/\lambda {)}^{k-1}{e}^{-(x/\lambda {)}^k}$ for $x\ge 0$ .

CDF

$F_X(x)=1-e^{-(x/\lambda {)}^k}$ for $x\ge 0$ .

Moments

$E[X]=\lambda \mathrm{\Gamma }(1+1/k)$ and $\mathrm{Var}(X)={\lambda }^2[\mathrm{\Gamma }(1+2/k)-\mathrm{\Gamma }(1+1/k{)}^2]$ .

Essential Result

When $k=1$ , Weibull $(1,\lambda )$ is exponential with rate $1/\lambda$ .

Small Example

If $k=1$ and $\lambda =5$ , then $E[X]=5$ , matching an exponential distribution with rate $0.2$ .

Common Mistakes

• Mixing Weibull scale with exponential rate.
• Assuming all Weibull models are memoryless; only $k=1$ is.

Connections

• Exponential Distribution
• Gamma Function
• Gamma Distribution

References