Problem 49 - Calculus for Biology and Medicine

Question

Suppose that the lifetime of a battery is exponentially distributed with an average life span of three months. What is the probability that the battery will last for more than four months?

Step-by-Step Solution

Answer

The probability that the battery lasts more than four months is approximately 0.7364.

Step 1: Identify Key Parameters

First, identify the key parameters in the problem statement. The average life span (mean) of the batteries is given as 3 months. For an exponential distribution, the mean equals the reciprocal of the rate parameter \( \lambda \). So, \( \lambda = \frac{1}{3} \text{ months}^{-1} \).

Step 2: Understand the Exponential Distribution

The exponential distribution is defined with the probability density function (PDF): \( f(x; \lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \). Here, the aim is to calculate the probability that a battery lasts more than 4 months, i.e., \( P(X > 4) \).

Step 3: Calculate the Probability

The probability that a battery lasts more than 4 months is given by the complement of the cumulative distribution function (CDF) at \( x = 4 \). Therefore, \( P(X > 4) = 1 - F(4) \), where \( F(x) = 1 - e^{-\lambda x} \).

Step 4: Apply CDF Formula

Substitute \( x = 4 \) and \( \lambda = \frac{1}{3} \) into the CDF formula: \( F(4) = 1 - e^{-\frac{1}{3} \times 4} = 1 - e^{-rac{4}{3}} \).

Step 5: Compute Numerical Value

Calculate \( e^{-rac{4}{3}} \) using a calculator, which is approximately 0.2636. Therefore, \( P(X > 4) = 1 - 0.2636 = 0.7364 \).

Step 6: Confirm the Result

Review each step and confirm the calculation to ensure that \( P(X > 4) = 0.7364 \) is accurate based on mathematical operations.

Key Concepts

Understanding Probability

Probability is a measure of the likelihood that a particular event will occur. It ranges between 0 and 1. A probability of 0 indicates that an event will not occur, whereas a probability of 1 indicates certainty of occurrence. In our battery problem, we are interested in the probability that a battery will last longer than four months. This involves understanding and applying probability concepts to an exponential distribution, a common model for time-to-failure analysis.

Probability can be expressed in different forms, including as a fraction, a decimal, or a percentage. Here, we calculate the probability as a decimal, reflecting the likelihood of a battery exceeding four months in its lifetime.

The Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is a fundamental concept in statistics. It describes the probability that a random variable will take a value less than or equal to a given point. In contrast to the probability density function (PDF), which provides probabilities over an infinitesimally small interval, the CDF accumulates these probabilities across ranges.

In the context of the exponential distribution, the CDF is given by the formula \( F(x) = 1 - e^{-\lambda x} \). This formula helps us compute the probability that a random event will occur up to a certain point. In our battery example, we use the CDF to calculate the probability of a battery not making it to 4 months and then subtract this from 1 to find the probability of lasting more than 4 months.

Rate Parameter (\( \lambda \))

The rate parameter, \( \lambda \), is a crucial part of the exponential distribution. It determines the rate at which the probability of failure increases. Specifically, it is the reciprocal of the mean or average lifespan in an exponential distribution.

• If the mean lifetime of a process or item, like our battery, is 3 months, then \( \lambda = \frac{1}{3} \text{ months}^{-1} \). This indicates a relatively slow failure rate, suggesting that batteries often function beyond their mean lifespan.
• Understanding \( \lambda \) is vital as it directly influences the shape of the exponential distribution curve. A higher \( \lambda \) suggests a steeper curve, indicating a rapid decrease in survival probability as time progresses.

Recognizing \( \lambda \) is the base for analyzing scenarios involving exponential distributions, such as estimating the reliability of batteries, machinery, or even electronic components.

Modeling Battery Lifetime with Exponential Distribution

Battery lifetime can often be modeled using the exponential distribution, which is particularly useful for quantifying time until an event occurs, like failure. This model assumes a constant failure rate, meaning that the probability a battery will fail in the next small interval of time does not change over the lifetime of the battery.

• The exponential distribution is memoryless, indicating that the lifetime expectancy of the battery does not change, regardless of how long it has already functioned. For instance, if a battery has lasted 4 months, the future expected failure curve remains the same as when it was new.
• In practical terms, this property helps predict manufacturing and warranty strategies, ensuring companies can provide accurate lifespan estimates based on statistical models.

By understanding how exponential distribution applies to battery lifetime, businesses and consumers alike can better anticipate battery performance and maintenance schedules.