The Weibull distribution generalizes the exponential distribution by introducing another parameter, \( b \), that allows for a varying hazard rate. It is typically used in reliability analysis and survival studies because it can model different forms of failure rates, including increasing, decreasing, or constant.
When the hazard rate is given by \( r(x) = c x^b \), where \( c \) and \( b \) are positive constants, it is indicative of a Weibull distribution. Applying this hazard rate into the definition \( \frac{f(x)}{1-F(x)} = c x^b \), we derive the probability density function:
\[ f(x) = c x^b e^{-\frac{c x^{b+1}}{b+1}} \]
For different values of \( b \), the shape of the distribution can vary:
- When \( b = 1 \), it simplifies to the exponential distribution.
- \( b < 1 \) indicates a decreasing failure rate.
- \( b > 1 \) suggests an increasing failure rate.
This flexibility makes the Weibull distribution highly adaptable to different types of life data.