Problem 118 - Applied Statistics and Probability for Engineers

Question

The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase a six-year-old automobile, with a working voltage regulator and plan to own it for six years. (a) What is the probability that the voltage regulator fails during your ownership? (b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure?

Step-by-Step Solution

Answer

(a) Probability of failure during ownership is approximately 0.6321. (b) Mean time until the next failure is 6 years.

Step 1: Understand the Exponential Distribution

An exponential distribution is characterized by its rate parameter \( \lambda \), which is the inverse of the mean. The mean given is 6 years, so \( \lambda = \frac{1}{6} \) per year.

Step 2: Calculate Probability of Failure in Six Years

The probability that an event occurs in time \( t \) for an exponential distribution is given by the survival function: \( S(t) = e^{-\lambda t} \). The probability of failure is \( 1 - S(t) \). For a 6-year period, \( \lambda = \frac{1}{6} \) and \( t = 6 \), so the probability of failure is:\[1 - S(6) = 1 - e^{-\frac{1}{6} \times 6} = 1 - e^{-1} \approx 0.6321\]

Step 3: Understand the Memoryless Property

Exponential distribution is memoryless, meaning that the future probability of failure depends only on the future time period, not on past duration.

Step 4: Mean Time to Next Failure After 3 Years

Once the voltage regulator fails and is replaced, the clock resets due to the memoryless property. The mean time to the next failure is again the mean of the exponential distribution, which is 6 years.

Key Concepts

Probability of Failure

When dealing with the exponential distribution in the context of reliability, "probability of failure" refers to the likelihood that a component fails within a certain time frame. In the given example, the life of a voltage regulator follows an exponential distribution with a mean of 6 years. The exponential distribution can effectively model this situation because it is well-suited for describing time between independent events, like failures.
To calculate the probability of failure during a specific time frame, we use the complementary cumulative distribution function (CCDF), also known as the survival function. It is expressed as:

• The probability that a component does not fail by time \( t \) is given by: \( S(t) = e^{-rac{1}{6}t} \).
• The probability of failure within time \( t \) is: \( 1 - S(t) \).

For example, if you intend to use the voltage regulator for an additional 6 years, the probability it will fail during this period is \( 1 - e^{-rac{1}{6} imes 6} \), which simplifies to approximately 0.6321, or 63.21%.
Remember, in exponential distributions, the longer you assess the time frame, the higher the probability of failure due to the decreasing survival function.

Mean Time to Failure

The Mean Time to Failure (MTTF) is a critical metric used in reliability engineering, especially when utilizing exponential distribution. It represents the average expected lifespan of a non-repairable system before it fails for the first time. For systems described by the exponential distribution, MTTF is simply the mean of the distribution.
For our voltage regulator, the MTTF is 6 years. This means, on average, across numerous units, each voltage regulator is expected to last 6 years before failure. It’s important to differentiate between mean and median in exponential distributions. Here, the mean time to failure provides a straightforward indication of the reliability over time.
The MTTF is constant, regardless of any previous life the component has had. This characteristic ties into the memoryless property, which plays a critical role in understanding future failure probabilities without concern for past operation durations.

Memoryless Property

One of the unique characteristics of the exponential distribution is its "memoryless property." This distinguishing property indicates that, for an exponential distribution, the probability of failure in the future is not dependent on the past, only the remaining time.

• This means the component does not "age"; at any point in time, its remaining life expectancy remains constant.
• Mathematically, for an exponential random variable \( X \), it states: \( P(X > s + t \mid X > t) = P(X > s) \).

In simpler terms, if you've already used a component for a certain period and it didn't fail, its expected future lifetime remains the same as though you just installed it. For example, if after using a voltage regulator for 3 years it fails and is replaced, the new regulator again has an average expected lifespan of 6 years. This property ensures that each new component starts fresh, with its reliability unaffected by any prior use.