Problem 119 - Applied Statistics and Probability for Engineers

Question

Suppose that the time to failure (in hours) of fans in a personal computer can be modeled by an exponential distribution with \(\lambda=0.0003 .\) (a) What proportion of the fans will last at least 10,000 hours? (b) What proportion of the fans will last at most 7000 hours?

Step-by-Step Solution

Answer

Around 4.979% of fans will last at least 10,000 hours, and about 87.754% will last at most 7,000 hours.

Step 1: Identify the Distribution and Key Formula

The time to failure is modeled by an exponential distribution with rate \( \lambda = 0.0003 \). The exponential distribution's survival function, which gives the probability of lasting at least \( t \) hours, is \( S(t) = e^{-\lambda t} \).

Step 2: Calculate Survival Probability for 10,000 Hours

To find the proportion of fans lasting at least 10,000 hours, use the survival function: \( S(10,000) = e^{-0.0003 \times 10,000} \). Simplifying inside the exponent gives \( e^{-3} \). Compute \( e^{-3} \) to get the proportion.

Step 3: Evaluate the Exponential Function

Calculate \( e^{-3} \) using a scientific calculator or logarithm tables to determine the exact numerical value, which is approximately \( 0.04979 \).

Step 4: Infer the Long-Lasting Proportion

Since \( 0.04979 \) is the result of the survival function, we can infer that about 4.979% of the fans will last at least 10,000 hours.

Step 5: Calculate Cumulative Distribution for 7,000 Hours

For part (b), calculate the proportion of fans that last at most 7,000 hours using the cumulative distribution function, \( F(t) = 1 - e^{-\lambda t} \). Thus, find \( F(7,000) = 1 - e^{-0.0003 \times 7,000} \).

Step 6: Evaluate the Exponent

Inside the exponent, calculate \( -0.0003 \times 7,000 = -2.1 \). Now compute \( 1 - e^{-2.1} \).

Step 7: Calculate the Exact CDF Value

Evaluate \( e^{-2.1} \) which is approximately \( 0.12246 \). Thus, \( 1 - e^{-2.1} = 0.87754 \).

Step 8: Infer the Short-Lasting Proportion

Thus, about 87.754% of the fans will last at most 7,000 hours.

Key Concepts

Survival Function

The survival function is a key concept when dealing with exponential distributions. It tells us the probability that a random variable, like the lifespan of a computer fan, survives past a certain time \( t \). Using the survival function allows us to calculate how long a product will last beyond a specified time. In this example, the function is given by:

• \( S(t) = e^{-\lambda t} \)

Here, \( \lambda \) is the rate parameter, which defines the distribution. In our case, it is \( 0.0003 \).
To find out what proportion of fans last at least 10,000 hours, we simply substitute \( t = 10,000 \) into the survival function. This gives us \( e^{-3} \), which means that about 4.979% of the fans remain operational after 10,000 hours. This result is found using a scientific calculator or logarithm tables, showing the power of understanding the survival function.

Cumulative Distribution Function

The cumulative distribution function (CDF) is another essential tool in probability and statistics. It helps us understand the probability that a random variable takes on a value less than or equal to a certain threshold. For the exponential distribution, the CDF is defined as:

• \( F(t) = 1 - e^{-\lambda t} \)

This function tells us how likely it is that an event, like fan failure, occurs within a specified timeframe.
Applying this to our problem, we find out how many fans last at most 7,000 hours. Plugging \( t = 7,000 \) into the CDF gives us \( 1 - e^{-2.1} \). Calculating, we find approximately 0.87754, or 87.754%. This means a large proportion of the fans are expected to fail before reaching 7,000 hours of operation. The CDF captures this understanding in a single formula, summarizing a lot of complex interactions neatly.

Exponential Model

The exponential model is frequently used to describe time-to-failure data, especially for products like computer fans. In this context:

• It provides a mathematical framework to model the time until an event occurs.
• The model assumes that events happen independently and continuously at a constant average rate, \( \lambda \).

For our fans, \( \lambda = 0.0003 \) means that on average, 0.03% of the fans fail per hour. The exponential model is powerful because it's both simple and widely applicable where time-to-failure data is concerned.
The exponential distribution is memoryless, meaning the probability of failure in the next moment doesn't depend on past events. This property simplifies many calculations and makes predictions about future failures straightforward. By using the exponential model in our exercise, we can make informed decisions about product lifetime and reliability.