Final answer:
The probability distribution for the time of completion is exponential with parameter λ. The expected time of completion is 1/λ. The probability of completing the task before 12 p.m. can be found using the cumulative distribution function.
Step-by-step explanation:
a) The probability distribution for the time of completion can be described using the exponential distribution with parameter λ. Since the duration of the task is exponentially distributed with parameter λ, the probability density function (PDF) for the time of completion, denoted by f(t), is given by f(t) = λe^(-λt), where t is the time of completion.
b) To calculate the expected time of completion, we need to find the mean of the exponential distribution. The mean, denoted by E(X), is given by E(X) = 1/λ. In this case, the expected time of completion is 1/λ.
c) To assess the probability of completing the task before 12 p.m., we need to find the cumulative distribution function (CDF) for the exponential distribution. The CDF, denoted by F(t), gives the probability that the time of completion is less than or equal to t. The probability of completing the task before 12 p.m. is given by F(12).
d) To analyze the reliability of the task completion process, we can use the survival function, denoted by S(t), which gives the probability that the time of completion is greater than t. The reliability of the task completion process can be assessed by calculating the probability of completing the task within a specified time frame, such as within 1 hour or within 30 minutes.