Office Manager Has Several Computers Running Distributed

An Office Manager Has Several Computers Running Distributed Programs

An office manager has several computers running distributed programs. Because of the demands on the system, the machines may crash at various times during the day and require a hard reset. The probability distribution for the number of times a randomly selected machine crashes during a day, X, is given in the table. Fill in the blanks.

1. The mean for the number of crashes by a single machine during a day is _(Answer 1)_. (Give your answer to two decimal places.)

2. The variance for the number of crashes by a single machine during a day is _(Answer 2)_. (Give your answer to four decimal places.)

3. The standard deviation for the number of crashes by a single machine during a day is _(Answer 3)_. (Give your answer to two decimal places.)

4. Suppose n = 2 machines are selected at random and the statistic T represents the total number of crashes for the two machines. Then the mean of T is _(Answer 4)_.

5. The variance of T is _(Answer 5)_. (Give your answer to four decimal places.)

6. The standard deviation of T is _(Answer 6)_. (Give your answer to four decimal places.)

Paper For Above instruction

The distribution of crashes among computers in an office environment is a pertinent topic in understanding system reliability and maintenance needs. This paper analyzes the probability distribution of crashes per machine per day, calculates key statistical measures for the distribution, and extends the analysis to multiple machines to understand the overall system behavior.

Suppose the probability distribution for the number of crashes, X, experienced by a single computer during a day can be summarized through a discrete probability table, which includes the probabilities associated with each possible number of crashes. Although the exact table isn't provided here, typical distributions might include values for crashes = 0, 1, 2, etc., with corresponding probabilities summing to 1. From this distribution, the mean, variance, and standard deviation can be calculated, which are fundamental metrics in probability and statistics for understanding the average behaviour and variability in crash counts.

The mean number of crashes, often denoted as E[X], reflects the average crashes per machine and is calculated as the sum of each number of crashes multiplied by its probability. The variance Var[X], measures the variability of crashes around this mean, and the standard deviation is the square root of the variance, providing a measure in the original units of crash counts. Once these measures are established, the analysis extends to the collective behavior of multiple machines. For n = 2 machines, the total crashes, T, are the sum of individual crash counts, assuming independence between machines. Under this assumption, the expected total crashes are twice the mean of a single machine, and the variance of the total is twice the variance of a single machine, owing to the properties of the sum of independent random variables.

Understanding the distribution and variability of system crashes helps in planning for maintenance, backup strategies, and system upgrades. It also supports risk assessments in large-scale computing environments, ensuring operational continuity. The calculations of mean, variance, and standard deviation form the backbone of this predictive capacity, while the extension to multiple machines provides insights into aggregate risks and system reliability.

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