Problem 61 - Calculus for Biology and Medicine

Question

Suppose the lifetime of a technical device is exponentially distributed with parameter \(\lambda=0.2 /\) year (a) What is the expected lifetime? (b) The median lifetime is defined as the age \(x_{m}\) at which the probability of not having failed by age \(x_{m}\) is \(0.5\). Find \(x_{m}\).

Step-by-Step Solution

Answer

The expected lifetime is 5 years, and the median lifetime is approximately 3.466 years.

Step 1: Understanding the Exponential Distribution

The lifetime of a technical device is described by an exponential distribution, which is a continuous probability distribution. It is often used for modeling the time until an event occurs, such as failure. The parameter \( \lambda \) (lambda) is the rate parameter of the distribution.

Step 2: Calculate the Expected Lifetime

For an exponential distribution with parameter \( \lambda \), the expected lifetime (mean) is given by the formula \( \frac{1}{\lambda} \). Given \( \lambda = 0.2 \), the expected lifetime is \( \frac{1}{0.2} = 5 \) years.

Step 3: Define the Median Lifetime

The median lifetime \( x_m \) is the value such that the probability of the device lasting longer than \( x_m \) is 0.5. Mathematically, this is the value where the survival function \( S(x_m) = 0.5 \).

Step 4: Solve for the Median Lifetime

The survival function \( S(x) = e^{-\lambda x} \). We set \( e^{-0.2 x_m} = 0.5 \) to find the median lifetime. Solving, we take the natural logarithm of both sides: \[\ln(e^{-0.2 x_m}) = \ln(0.5)\]This simplifies to:\[-0.2 x_m = \ln(0.5)\] Thus, \[x_m = -\frac{\ln(0.5)}{0.2} \approx 3.466\]Therefore, the median lifetime is approximately 3.466 years.

Key Concepts

Expected Lifetime

The expected lifetime of a device is a useful measure in ensuring reliability and planning maintenance. When dealing with an exponential distribution, this expected value helps us understand how long a device can be expected to operate before it fails. For an exponential distribution with rate parameter \( \lambda \), the expected lifetime is calculated as the reciprocal of the rate parameter. Specifically:

• Expected lifetime = \( \frac{1}{\lambda} \)

For a rate \( \lambda = 0.2 \text{ per year} \), the calculation is straightforward:\[\text{Expected Lifetime} = \frac{1}{0.2} = 5 \text{ years}\]This result tells us that, on average, the device is expected to last 5 years before failure. This assumes a constant failure rate over time, characteristic of the exponential distribution.

Understanding this concept further allows the prediction of large-scale device behavior in industries, promoting efficient planning and resource allocation.

Survival Function

The survival function is a critical concept in reliability analysis, as it indicates the probability of a device surviving beyond a certain time, \( x \). For the exponential distribution, this function is represented as:\[S(x) = e^{-\lambda x}\]This equation tells us how the probability of survival decreases over time. The survival function starts at 1 when \( x \) is 0 and asymptotically approaches 0 as \( x \) extends to infinity.

Because the function describes how the likelihood of survival decreases, it is invaluable for evaluating longevity and planning for device lifecycles. In practical terms, you can see how this function helps in determining maintenance schedules and preemptive replacements. If a specific survival probability is desired, the function can guide how long a device is expected to function properly without failure.

Median Lifetime

The median lifetime is the point at which there is a 50% chance that the device will still be in operation. It's an essential benchmark for assessing reliability, especially for product warranty considerations and performance guarantees. To find the median lifetime in an exponential distribution, we solve the equation:\[S(x_m) = 0.5\]Given the survival function formula \( S(x) = e^{-\lambda x} \), we set:\[ e^{-0.2 x_m} = 0.5\]By taking the natural logarithm of both sides and solving for \( x_m \):\[\ln(e^{-0.2 x_m}) = \ln(0.5)\]This simplifies to:\[-0.2 x_m = \ln(0.5)\]Hence,\[x_m = -\frac{\ln(0.5)}{0.2} \approx 3.466 \text{ years}\]In conclusion, the median lifetime of approximately 3.466 years means there is an equal probability that a device will last longer or shorter than this time. This interpretation of a device's median lifetime can guide decisions on product lifecycles and consumer expectations.