The average lifetime of a certain new cell phone is 3.4 years. The manufacturer will replace any cell phone failing within three years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution. 68% of these phone last how long (in years)?

Given that the lifetime of these cell phones follows an exponential distribution with a mean lifetime of 3.4 years, the goal is to find the range of lifetimes within which 68% of the phones will last.

### Step 1: Understanding the Exponential Distribution
The probability density function (PDF) of an exponential distribution is given by:

\[
f(x; \lambda) = \lambda e^{-\lambda x}
\]

where \( \lambda = \frac{1}{\mu} \), and \( \mu \) is the mean of the distribution. Here, the mean \( \mu \) is given as 3.4 years, so:

\[
\lambda = \frac{1}{3.4} \approx 0.2941
\]

### Step 2: Finding the Range for 68% of the Distribution
In the case of a normal distribution, the 68% range corresponds to one standard deviation away from the mean. For an exponential distribution, the standard deviation is equal to the mean \( \mu \).

To find the range that covers 68% of the phones, we will calculate the 16th percentile and the 84th percentile of the exponential distribution. This will give us the bounds for which 68% of the lifetimes fall.

The cumulative distribution function (CDF) of the exponential distribution is:

\[
F(x; \lambda) = 1 - e^{-\lambda x}
\]

We need to solve for \( x \) in the following equations:

- For the 16th percentile: \( F(x_{16\%}) = 0.16 \)
- For the 84th percentile: \( F(x_{84\%}) = 0.84 \)

### Step 3: Solving for the Percentiles
Using the CDF equation, we can solve for the values of \( x_{16\%} \) and \( x_{84\%} \).

Let me calculate these values.The range of lifetimes within which 68% of the cell phones will last is approximately between **0.59 years** and **6.23 years**.

### Summary:
- **16th percentile**: 0.59 years
- **84th percentile**: 6.23 years

This means that 68% of these cell phones are expected to have a lifetime within this range.

Would you like further details or have any questions?

### Related Questions:
1. How do you calculate the median lifetime for an exponential distribution?
2. What is the probability that a phone will last longer than 5 years?
3. How does the manufacturer’s replacement policy affect their expected costs?
4. How would the range change if the mean lifetime were 5 years instead of 3.4 years?
5. What percentage of phones last more than the average lifetime of 3.4 years?

### Tip:
For exponential distributions, the mean and standard deviation are equal, making them simpler to analyze when considering averages.

Learn how to calculate the range of lifetimes within which 68% of cell phones will last, given an exponential distribution with a mean lifetime of 3.4 years. Explore the probability density function (PDF) and cumulative distribution function (CDF) of exponential distributions. Suitable for advanced college students and researchers.

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