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Understanding how to calculate the mean lifetime of an individual in the context of a Poisson process is crucial for any student of statistics or probability. In the given problem, we're asked to assume that mistakes in cell division occur at a steady rate and lead to the death of an individual at a certain threshold.

To find the mean lifetime, which is the average time until the individual reaches 196 mistakes, we multiply the rate of mistakes by the time. With a rate of 2.5 mistakes per year, the equation becomes \(196 = 2.5 \times t\). Solving for \(t\) gives us 78.4 years, which is the mean lifetime according to this model. It's important to note that mean lifetime is an average; individuals may die sooner or live longer, but on average, the lifetime is 78.4 years.

The variance is a statistical measure that tells us how much the lifetime of individuals can differ from the mean. In a Poisson process, the variance is equal to its mean, which here is the number of events (mistakes) in that period. Since we've already established a rate of 2.5 mistakes per year and a mean lifetime of 78.4 years, the variance is \(2.5 \times 78.4 = 196\) years squared.

This number may seem large, but remember, variance is not expressed in the same units as the data itself. To understand the typical deviation from the mean, one would take the square root of the variance, which in statistics is known as the standard deviation.

The Cumulative Distribution Function (CDF) is a fundamental concept in the study of probability distributions, including the Poisson distribution. The CDF gives the probability that a Poisson-distributed random variable \(X\) is less than or equal to a certain value \(k\). Mathematically, it's the sum of probabilities for all events up to \(k\):

\( F(k; \lambda) = P(X \le k) = \sum_{i=0}^k \frac{e^{-\lambda}(\lambda)^i}{i!} \)

For practical calculations, especially when dealing with a large sum like in our case, using a calculator or statistical software can greatly simplify the process.

When dealing with large datasets or complex distributions, exact probabilities can be cumbersome to calculate. Approximations provide a more manageable approach. In our scenario, we use the CDF of the Poisson process to approximate probabilities of an individual dying before reaching certain ages.

To approximate the probability of dying before age 67.2 years, we calculate the CDF for 195 events, yielding approximately 0.4136. Similarly, the probabilities of reaching ages 90 and 100 can be estimated using 1 minus the CDF for the events leading up to these ages. These approximations allow for a much quicker assessment of the probabilities without requiring exhaustive calculations.

The Poisson distribution is an essential tool for modeling the probability of a given number of events happening in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event. The provided problem is a textbook example, where the occurrences of mistakes in cell division follow a Poisson process.

Understanding the properties of the Poisson distribution such as its mean, variance, and how to use its CDF is crucial for analyzing and interpreting data that fits this type of stochastic process. It's especially useful in various fields, ranging from biology and medicine to telecommunications and traffic engineering.