Final answer:
The calculation of variance requires the probability density function or additional statistical data. Without this information, we cannot provide the variance of the lifetime of the machine.
Step-by-step explanation:
The question revolves around determining the variance of the lifetime of a machine whose mean lifetime is given as 12 years. Variance is a measure of how data points differ from the mean value, indicating the spread or dispersion within a dataset.
The provided options suggest that this is a multiple choice question with potential variance values. However, without the actual probability density function or additional statistical data regarding the distribution of the lifetimes, we cannot calculate or accurately predict the variance.
Variance can usually be calculated using the formula σ^2 = E[(X - μ)^2], where σ^2 is the variance, E is the expectation operator, X is the random variable, and μ is the mean of the random variable. Empirical data or a more complete description of the probability density function is essential to provide a meaningful answer to this question.