Final answer:
In the given statistical problems, exponential distribution is used to calculate various probabilities for the life expectancy of LED bulbs, such as the likelihood of a bulb lasting less than a certain duration or the warranty period based on LED longevity.
Step-by-step explanation:
The question relates to the normal distribution and exponential distribution of the life expectancy of LED bulbs, modeled using probability theory and the properties of these distributions. Specifically, the focus is on calculating probabilities associated with life spans of LED bulbs and exploring related theoretical warranty periods.
- To find the probability that a LED bulb lasts less than one year, use the cumulative distribution function of the exponential distribution where the mean lifetime is eight years.
- The probability of a bulb lasting between six and 10 years is also found through the exponential distribution function.
- Life expectancy calculations help determine the time period where 70% of all bulbs last at least a certain number of years.
- For warranty considerations, using the 2nd percentile helps define the cutoff lifetime.
In the context of a bulb lasting beyond 12 years, the probability it would last more than a total of 19 years can be calculated using the same distribution.