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Chapter 5: Problem 36

Suppose that the life distribution of an item has the hazard rate function \(\lambda(t)=t^{3}, t>0 .\) What is the probability that (a) the item survives to age \(2 ?\) (b) the item's lifetime is between .4 and \(1.4 ?\) (c) a 1-year-old item will survive to age \(2 ?\)

Short Answer

Expert verified
(a) The probability that the item survives to age \(2\) is approximately \(1.83\%\). (b) The probability that the item's lifetime is between \(0.4\) and \(1.4\) is approximately \(75.20\%\). (c) The probability that a 1-year-old item will survive to age \(2\) is approximately \(2.35\%\).

Step by step solution

01

1. Calculate Cumulative Hazard Function (Integration)

In order to find the survival function, we first need to calculate the cumulative hazard function, \(H(t)\). \(H(t)\) is the integral of the hazard rate function \(\lambda(t)\), i.e., \(H(t) = \int_0^t \lambda(u) \, du\). Since \(\lambda(t) = t^3\), we find the cumulative hazard function by integrating with respect to \(u\): \[H(t) = \int_0^t u^3 \, du\]
02

2. Calculate the Survival Function (Exponential)

Now that we have the cumulative hazard function, we can calculate the survival function, \(S(t)\), using the exponential function: \[S(t) = e^{-H(t)}\] Substitute the cumulative hazard function, \(H(t)\), from step 1 into the equation: \[S(t) = e^{-\int_0^t u^3 \, du}\] Now, we will first solve the integral part in the exponent. \[\int_0^t u^3 \, du = \left[\frac{1}{4}u^4\right]_0^t = \frac{1}{4}t^4\] Now, plug this result back into the survival function equation. \[S(t) = e^{-\frac{1}{4}t^4}\]
03

3. Calculate the Probabilities Using the Survival Function

(a) To find the probability that the item survives to age 2, evaluate the survival function \(S(t)\) at \(t=2\): \[S(2) = e^{-\frac{1}{4}(2^4)}=e^{-4} \approx 0.0183\] So, there is approximately an 1.83% chance that the item survives to age 2. (b) To find the probability that the item's lifetime is between 0.4 and 1.4, calculate the differences in the survival function values at these points: \[P(0.4 < t < 1.4) = S(0.4) - S(1.4)\] \[S(0.4) = e^{-\frac{1}{4}(0.4^4)} \approx 0.9830\] \[S(1.4)= e^{-\frac{1}{4}(1.4^4)} \approx 0.2310\] \[P(0.4 < t < 1.4) = 0.9830 - 0.2310 \approx 0.7520\] So, there is approximately a 75.20% chance that the item's lifetime is between 0.4 and 1.4. (c) To find the probability that a 1-year-old item will survive to age 2, we find the conditional probability of survival given that the item is already 1-year-old, which is \(\frac{S(2)}{S(1)}\). We already calculated \(S(2)\), so we just need to calculate \(S(1)\): \[S(1) = e^{-\frac{1}{4}(1^4)}=e^{-\frac{1}{4}} \approx 0.7788\] Then, the conditional probability is: \[\frac{S(2)}{S(1)} = \frac{e^{-4}}{e^{-\frac{1}{4}}} \approx 0.0235\] So, there is approximately a 2.35% chance that a 1-year-old item will survive to age 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hazard Rate Function
The hazard rate function, often simply termed as the hazard function, is a crucial element in survival analysis. It is a non-negative function, \(\lambda(t)\), which describes the instantaneous rate of failure at time \(t\), assuming the item has survived up until time \(t\). In other words, it is a way of quantifying the risk that an item will fail at a specific moment in time, given that it has not failed yet.

For an item whose life distribution follows \(\lambda(t) = t^3\), we're looking at a hazard rate that increases with time, suggesting that the older the item, the greater the risk of failure. This can be interpreted as a sign that wear and tear or other age-related factors increase the likelihood of the item's failure as time progresses.

Understanding the hazard rate function is crucial in survival analysis because it lays the groundwork for determining other important functions, such as the cumulative hazard function and the survival function. By integrating the hazard rate function, you derive the cumulative hazard function, which is fundamental to calculating the survival probability of an item over time.
Cumulative Hazard Function
The cumulative hazard function, denoted as \(H(t)\), aggregates risk over time. It is derived by integrating the hazard rate function. In essence, \(H(t)\) provides the accumulated risk up until a specific time \(t\). This function reflects the total experience of risk an item has encountered, making it a foundational component in survival analysis.

For our hazard rate \(\lambda(t) = t^3\), the cumulative hazard function is obtained by integrating: \[ H(t) = \int_0^t u^3 \, du = \frac{1}{4}t^4 \]
This integration tells us that the cumulative hazard grows as a quartic function of time, exponentially accumulating risk as time progresses. As risk collects, it serves to diminish the chances of survival over extended periods.

Calculating \(H(t)\) is crucial because it directly feeds into the survival function, which models the probability of an item surviving past a certain age. It translates the notion of risk at any given point into an overall measure of survival.
Survival Function
The survival function, usually represented by \(S(t)\), articulates the probability that an item will persist without failure past a certain time \(t\). It is mathematically linked to the cumulative hazard function using the exponential function: \[ S(t) = e^{-H(t)} \]
This equation suggests that as the cumulative hazard \(H(t)\) increases, the survival probability decreases, embodying the intuition that accumulated risk negatively impacts an item's likelihood of continued function.

In our example, the survival function follows as: \[ S(t) = e^{-\frac{1}{4}t^4} \]
This reflects how the exponential accumulation of risk (a result of \(t^3\)) impacts the item's chance of surviving over time. As we calculated, different values of the survival function at specific times, such as \(S(2) = e^{-4}\), offer clear, quantitative probabilities of survival.
  • At \(t = 2\), there's about a 1.83% chance of survival.
  • For an interval like 0.4 to 1.4, survival probability can be found by subtracting: \( S(0.4) - S(1.4) \).
  • The conditional survival for a one-year-old item till the second year, found as \(\frac{S(2)}{S(1)}\), shows the updated probability given it's already survived a year.
The survival function is a cornerstone of survival analysis, providing a clear perspective on an item's life expectancy and risk management.

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