Final answer:
To find the expected number of failures in 1 year of operation, multiply the hazard function by the time in operation. The expected value E(X) for the number of reports a safety engineer needs to read before finding an accident report caused by employee failure to follow instructions is approximately 2.86, based on a success probability of p = .35 under a geometric distribution.
Step-by-step explanation:
The question is asking for the expected number of failures in 1 year of operation, given a hazard function estimate.
To find the expected number of failures, we need to determine the rate of failure and multiply it by the time in operation. In this case, the hazard function represents the failure rate. The expected number of failures can be calculated by multiplying the hazard function (p) by the time in operation (1 year).
Expected number of failures = hazards function * time in operation = p * 1 year
The expected value E(X) for the number of reports a safety engineer needs to read before finding an accident report caused by employee failure to follow instructions is approximately 2.86, based on a success probability of p = .35 under a geometric distribution.
Geometric Random Variable and Expected Value Calculation
The question revolves around a geometric random variable, where we are interested in finding the number of trials a safety engineer must examine to find a report that indicates an accident caused by employee failure to follow instructions. With the probability of success (finding such a report) given as p = .35, we can calculate the expected value E(X), also known as the mean (μ), using the formula for the expected value of a geometric distribution: E(X) = 1/p. Therefore, E(X) = 1/.35, which calculates to approximately 2.86. This represents the average number of reports the safety engineer would need to read before finding one caused by failure to follow instructions.
The expected number of failures or defects in other settings, such as the expected number of defective products in a factory or the expected number of accidents in a city with a Poisson distribution, can similarly be predicted using appropriate statistical models and the given probabilities.