Problem 198 - Applied Statistics and Probability for Engineers

Question

The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with \(\lambda=0.00004 .\) What is the probability that the time until failure is (a) At least 20,000 hours? (b) At most 30,000 hours? (c) Between 20,000 and 30,000 hours?

Step-by-Step Solution

Answer

(a) 0.4493; (b) 0.6988; (c) 0.1481.

Step 1: Understanding the Problem

The time to failure follows an exponential distribution with rate parameter \(\lambda = 0.00004\). We need to find probabilities related to the distribution at various time intervals.

Step 1: Probability of Failure after a specific time period

For an exponential distribution with rate \(\lambda\), the probability that a random variable \(X\) is greater than \(t\) is given by \(P(X > t) = e^{-\lambda t}\). To find \(P(X \geq 20000)\), we calculate \(e^{-0.00004 \times 20000}\).

Step 3: Calculating Step 1

Substitute the values: \(P(X \geq 20000) = e^{-0.8} \approx 0.4493\). Therefore, the probability that the time until failure is at least 20,000 hours is \(0.4493\).

Step 2: Probability of Failure within a specific time period

The probability that \(X\) is less than or equal to \(t\) is \(P(X \leq t) = 1 - e^{-\lambda t}\). To find \(P(X \leq 30000)\), compute \(1 - e^{-0.00004 \times 30000}\).

Step 5: Calculating Step 2

Substitute the values: \(P(X \leq 30000) = 1 - e^{-1.2} \approx 0.6988\). Thus, the probability of failure at most 30,000 hours is approximately \(0.6988\).

Step 3: Probability of Failure between two time periods

To find the probability that \(X\) is between 20,000 and 30,000 hours, use the formula: \(P(20000 < X < 30000) = P(X \leq 30000) - P(X \leq 20000)\).

Step 7: Calculating Step 3

Using earlier calculations: \(P(X \leq 20000) = 1 - e^{-0.8} \approx 0.5507\). Therefore, \(P(20000 < X < 30000) = 0.6988 - 0.5507 = 0.1481\). Hence, the probability that failure occurs between 20,000 and 30,000 hours is about 0.1481.

Key Concepts

Probability Calculation

Calculating the probability when dealing with an exponential distribution often involves determining the likelihood of an event happening within specific constraints. For the context of failure time, these calculations tell us how likely it is for a device to operate without failing for a given period. When a rate parameter, \(\lambda\), is applied to an exponential distribution, it helps in determining probabilities using its unique formula. The key equations to calculate probabilities include:

• For times greater than a certain point: \(P(X > t) = e^{-\lambda t}\)
• For times less than or equal to a certain point: \(P(X \leq t) = 1 - e^{-\lambda t}\)

Learn to substitute values for \(\lambda\) and \(t\) directly into these equations for easy and effective results.
Calculating these can help predict how systems behave over time, especially how likely failure or success in a process is under random conditions.

Failure Time Analysis

Failure time analysis is crucial for understanding when devices or systems may break down in real-world applications. Specifically, it helps in planning maintenance, assessing risk, and designing systems that are more resilient. In the exercise above, the analysis indicates how many hours a laser in a cytometry machine will function before it is likely to fail.
Using the exponential distribution model, one can determine, for example, that the laser will function for at least 20,000 hours with a calculated probability. This probability is derived by applying the formula for the exponential distribution provided above. By determining these probabilities, businesses and scientists can better manage resources and make informed decisions about equipment maintenance and replacements.
Failure time analysis thus transforms raw probability into meaningful data which organizations can use to improve their operational efficiency.

Probability Distribution Function

In any probability study involving exponential distribution, the Probability Distribution Function (PDF) is your primary tool. This function helps us determine the likelihood of a random variable taking on certain values. For exponential distributions, which typically model the time between events in a Poisson process, the PDF is defined as:\[ f(x; \lambda) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0 \]This PDF showcases the memoryless property of exponential distributions, which means that the probability of an event occurring in the future is independent of past events.
Understanding the PDF provides a clearer picture of how time-to-failure or similar random processes distribute over time. When interpreting an exercise, this function explains the mechanics behind probability equations and the outcomes. It helps predict where values like time-to-failure will fall, and underlies all probability calculations performed in step-by-step solutions.