Problem 28 - An Introduction to Mathematical Statistics and Its Applications

Question

Fifty spotlights have just been installed in an outdoor security system. According to the manufacturer's specifications, these particular lights are expected to burn out at the rate of \(1.1\) per one hundred hours. What is the expected number of bulbs that will fail to last for at least seventy-five hours?

Step-by-Step Solution

Answer

The expected number of bulbs that will fail to last for at least seventy-five hours is approximately 41.

Step 1: Calculate the Expected Rate of Failure for Seventy-Five Hours

The rate of failure given is for 100 hours. We need to find the rate for 75 hours. To accomplish this, we can use a proportion: if \(1.1\) bulbs fail per \(100\) hours, we can denote \(x\) as the number of bulbs that would fail per \(75\) hours. Hence, we get this proportion equation: \(1.1/100 = x/75\). Solving for \(x\), gives \(x = (1.1 * 75) / 100 = 0.825\). So the expected failure rate for 75 hours is \(0.825\) bulbs.

Step 2: Apply the Poisson Distribution

The Poisson distribution is useful for calculating the probability of a number of events in a fixed interval of time or space. The formula for the Poisson distribution is: \( P(X = k) = (λ^k * e^{-λ}) / k! \), where λ is the expected number of occurrences, k is the actual number of occurrences, and e is the base of the natural logarithm. Applying the Poisson distribution to 50 bulbs, we would have λ = 50*0.825 = 41.25.

Step 3: Calculate the Expected Number of Bulbs Failing

The expected value E(X) of a Poisson random variable X (which is the number of bulbs failing) is equal to the parameter λ. So, the expected number of bulbs that will fail in 75 hours is 41.25.

Key Concepts

Expected Rate of Failure

In any statistical analysis involving probabilities of failures, the expected rate of failure is a crucial concept. For this specific exercise, we start with the manufacturer's failure rate, which is provided per 100 hours.To find out the expected rate of failure for a different time period, here 75 hours, we use a simple proportion equation.

• We know that 1.1 bulbs are expected to fail per 100 hours.
• We set up a proportion to find the failure rate for 75 hours: \( \frac{1.1}{100} = \frac{x}{75} \).

Solving for \(x\) involves basic arithmetic:\[x = \frac{1.1 \times 75}{100} = 0.825\]This means, approximately 0.825 bulbs are expected to fail every 75 hours.Using proportions allows us to easily determine expected rates over shorter or different time periods by scaling down the larger intervals provided by manufacturers.

Outdoor Security System

Outdoor security systems, such as the one described here, heavily rely on consistent performance from their components. The system mentioned consists of fifty spotlights. The longevity and reliability of each spotlight directly impact the overall effectiveness of the security system.

• Having a grasp on how often these lights might fail helps in planning maintenance and replacements.
• It enables security system managers to predict when and how often to check the systems to ensure they're always operational.
• This minimizes unexpected dark spots that might compromise the security.

Understanding the expected rate of failure helps in the logistical management of such systems, ensuring that they serve their intended purpose without frequent outages.

Proportion Equation

The use of proportion equations is an essential mathematical tool in calculating expected outcomes based on known rates. In this exercise, proportions help bridge the manufacturer's failure rate over 100 hours to our required time span of 75 hours.The process involves:

• Setting up a fraction representing the known rate of failure (per 100 hours),
• Equaling it to a fraction representing the unknown failure rate for 75 hours.

For example:\[\frac{1.1}{100} = \frac{x}{75}\]Solving for \(x\) involves cross-multiplying and dividing, which yields:\[x = \frac{1.1 \times 75}{100} = 0.825\]This scenario illustrates how versatile proportion equations can be, as they allow for flexible adjustments in expected rates across various timescales.