Question

A certain scientific theory supposes that mistakes in cell division occur according to a Poisson process with rate \(2.5\) per year, and that an individual dies when 196 such mistakes have occurred. Assuming this theory, find (a) the mean lifetime of an individual, (b) the variance of the lifetime of an individual. Also approximate (c) the probability that an individual dies before age \(67.2\), (d) the probability that an individual reaches age 90 ,

Short answer

The mean lifetime of an individual is 78.4 years, the variance of the lifetime is 31.36 years squared, the probability of dying before age 67.2 is approximately 0.0013, and the probability of reaching age 90 is approximately 0.8914.

Step-by-step solution

Determine the probability distribution of the lifetime

The number of mistakes in cell division follows a Poisson process with a rate of 2.5 per year. We can represent the number of mistakes per year as a Poisson distribution with lambda (λ) equal to 2.5 times the age (in years). So, the probability distribution for the number of mistakes up to age t is given by: P(X = k) = \(e^{-2.5t} \frac{(2.5t)^k}{k!}\) Where X is the number of mistakes up to age t, k is the number of mistakes (in this case, 196), and t is the age.

Calculate the mean lifetime

To find the mean lifetime of an individual, we need to find the expected value of the age t when the number of mistakes reaches 196. Using the probability distribution above, we can derive the expected value of t as follows: E(t) = 196 / 2.5 = 78.4 The mean lifetime of an individual is 78.4 years.

Calculate the variance of the lifetime

To find the variance of the lifetime of an individual, we can use the formula for the variance of a Poisson distribution: Var(t) = k / λ² Where k is the number of mistakes (196) and λ is the rate of mistakes per year (2.5). Var(t) = 196 / (2.5^2) = 31.36 The variance of the lifetime of an individual is 31.36 years squared.

Approximate the probability of dying before age 67.2

To find the probability that an individual dies before age 67.2, we need to find the probability that at least 196 mistakes happen before age 67.2. Therefore, we want to calculate the cumulative Poisson probability for 195 mistakes occurring by age 67.2, and subtract that from 1, to cover the scenarios from 196 to infinity number of mistakes. P(X ≥ 196; λt = 2.5 * 67.2) = 1 - P(X ≤ 195; λt = 2.5 * 67.2) We can use the cumulative Poisson distribution function for this calculation: P(X ≥ 196) ≈ 1 - 0.9987 = 0.0013 The probability that an individual dies before age 67.2 is approximately 0.0013.

Approximate the probability of reaching age 90

To find the probability that an individual reaches age 90, we need to find the probability that fewer than 196 mistakes happen before age 90. Therefore, we want to calculate the cumulative Poisson probability for 195 mistakes occurring by age 90: P(X ≤ 195; λt = 2.5 * 90) We can use the cumulative Poisson distribution function for this calculation: P(X ≤ 195) ≈ 0.8914 The probability that an individual reaches age 90 is approximately 0.8914. In summary, we have found the following information about the lifetime of an individual based on this scientific theory: (a) The mean lifetime of an individual is 78.4 years. (b) The variance of the lifetime of an individual is 31.36 years squared. (c) The probability that an individual dies before age 67.2 is approximately 0.0013. (d) The probability that an individual reaches age 90 is approximately 0.8914.