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 will last more than four months is approximately 0.2636.

Step 1: Understand the Exponential Distribution

The problem states that the lifetime of the battery follows an exponential distribution with an average (mean) life span of 3 months. In an exponential distribution, the rate parameter \( \lambda \) is the reciprocal of the mean, which is calculated as \( \lambda = \frac{1}{ ext{mean}} \).

Step 2: Calculate the Rate Parameter \( \lambda \)

Given the mean lifetime \( \mu = 3 \) months, calculate \( \lambda \) using \( \lambda = \frac{1}{3} \). So, \( \lambda = \frac{1}{3} \) months.

Step 3: Formula for the Exponential Probability

The probability that a battery lasts longer than a time \( t \) is given by the cumulative distribution function: \( P(X > t) = e^{-\lambda t} \).

Step 4: Substitute Values into the Formula

We want to calculate the probability that the battery will last more than 4 months. Substitute \( t = 4 \) into the formula: \( P(X > 4) = e^{-\frac{1}{3} \, \times 4} \).

Step 5: Calculate the Exponential Expression

Evaluate the expression: \( P(X > 4) = e^{-\frac{4}{3}} \). This simplifies to \( e^{-\frac{4}{3}} \approx e^{-1.333333} \approx 0.2636 \).

Key Concepts

Probability Calculation

When dealing with the exponential distribution, a key calculation involves determining the likelihood of an event occurring after a certain period of time. In this context, we are trying to find out the probability that a battery will last longer than a specified time. The relevant formula, which uses the exponential probability distribution, is:

• \( P(X > t) = e^{-\lambda t} \)

Here, \( P(X > t) \) represents the probability that the event lasts more than time \( t \), \( \lambda \) is the rate parameter, and \( e \) is the base of the natural logarithm.
To calculate this, you need to substitute the correct values for \( \lambda \) and \( t \). In our problem, if we want to calculate the probability that a battery lasts more than 4 months, we substitute \( t = 4 \) and use the value of \( \lambda \) we have computed. When you evaluate the expression \( e^{-\lambda t} \), you will find this probability. It's a way to gauge how long beyond a certain point the battery will keep working.

Rate Parameter

The rate parameter \( \lambda \) is a fundamental concept in an exponential distribution. This parameter essentially represents the rate at which events occur. It's the reciprocal of the average or mean interval between events. In our example, the average lifetime of the battery is 3 months. Accordingly, the rate parameter is computed as \( \lambda = \frac{1}{3} \), which tells us that on average, events happen every 3 months.
Why is this important? Moving beyond definitions, understanding \( \lambda \) helps us predict and calculate probabilities related to time-based occurrences. The value of \( \lambda \) directly influences how quickly the probability decreases over time, impacting how we view the expected lifetimes of items like batteries.
By knowing \( \lambda \), we essentially get a sense of both the duration and urgency of processes modeled by exponential distributions. This helps us in making informed decisions based on statistical analysis.

Cumulative Distribution Function

In probability theory, the cumulative distribution function (CDF) is a crucial tool for understanding the distribution of values a random variable can take. For the exponential distribution, the CDF can be expressed in terms of the probability of an event happening beyond a certain point in time.
Here's the relevant formula for an exponential CDF:

• \( P(X > t) = e^{-\lambda t} \)

This formula shows the probability of a random variable being greater than a specific value \( t \), given a rate parameter \( \lambda \).
The exponential CDF is unique in its simplicity and exponential decay, representing how probabilities drop off sharply for longer durations. In practical terms, this means that if you're analyzing battery life, the exponential CDF tells you the likelihood of a battery lasting longer than a certain number of months. By examining the slope dictated by \( \lambda \), you gain insight into how probable it is for events to occur beyond any given period. This allows us to quantify expectations and make estimations grounded in statistical certainty, thus aiding both practical and theoretical decision-making.