91Ó°ÊÓ

The failure density function, denoted as \(f(t)\), is a crucial concept in reliability engineering and survival analysis. It represents the probability density of failure time for a component or system. To derive the failure density function, we use the given hazard function, \(h(t)\), and an exponential term involving the hazard function's integral. Here’s a step-by-step explanation:

• Start with the given hazard function \(h(t)\).
• Calculate its integral over the interval from 0 to \(t\).
• Substitute these results back to define \(f(t)\).

The formula for the failure density function is: \[ f(t) = h(t) \, \text{exp} \bigg[- \bigg( \frac{t^3}{3} - 2t^2 + 9t \bigg) \bigg] \] Here, \( \text{exp} \) refers to the exponential function, which naturally arises in decay processes. This equation integrates all vital elements, including the hazard function and its integral, to effectively represent the likelihood of failure at any given time \(t\).

Definite integrals play a critical role in calculating the area under a curve within a specific interval. In our context, we use definite integrals to evaluate the integral of the hazard function \(h(x)\) from 0 to \(t\). The steps involve:

• First, find the antiderivative (indefinite integral) of \(h(x) = x^2 - 4x + 9\).
• Then, evaluate this antiderivative at both the upper limit \(t\) and the lower limit (0).
• Subtract the value at the lower limit from the value at the upper limit to get the definite integral.

In mathematical terms, we compute: \[ \bigg[ \frac{x^3}{3} - 2x^2 + 9x \bigg]_0^t = \frac{t^3}{3} - 2t^2 + 9t \] This expression is integral to formulating the failure density function \(f(t)\), as it is used in the exponential term.

The hazard rate, or hazard function \(h(t)\), quantifies the instantaneous failure rate at any given time \(t\). For the given problem, the hazard function is: \[ h(t) = t^2 - 4t + 9 \ (0 < t \, \leq \, 1) \] To compute the failure density function, we must:

• Understand the behavior of \(h(t)\) for the given interval.
• Integrate \(h(t)\) over the desired range to use in exponential decay calculations.
• Combine \(h(t)\) with it’s integral in the failure density function formula.

The hazard rate provides a foundation for further analyses, helping us understand how likely a component is to fail at a specific time instant. For our specific hazard function, the failure density function incorporates an exponential term influenced by the integral of \(h(t)\), showing how integrated hazard rates impact overall failure probabilities. In summary, the hazard rate helps predict the failure behavior over time, proving indispensable for both theoretical and practical applications in reliability engineering.