I am having trouble with a series of system tests in which each test is assumed to be a normal approximations of binomial distribution. Each test sample, n, is a Bernoulli ("success", "failure") trial of the system. Each system test has approximately 550 IID observations. Each has a lifetime that may vary between 2-8 seconds, regardless of success or failure. During this lifetime a success or failure will occur. Therefore, for each system test an estimated probability of failure, $p$, is determined where $p = \rm{Pr}[X=0]$.

However, since the test data in which the estimated probability of failure was determined is available, I wish to develop a CDF, F(t), that provides the probability of failure per unit time (0.1 seconds). More specifically, I wish to develop a graph to show when the failure occurred.

I have a few questions that I am having problems with:

  1. I would like to develop the CDF's, but I am not sure whether the lifetime differences of each sample varying from 2-8 seconds may bias the CDF curve. This curve will provide the cumulative estimated probability of failure in 0.1 second increments.

  2. Do the number of failures within each time interval affect the significance of the estimated probability of failure within that particular time interval?

  3. In addition I would like to develop two corresponding curves that illustrate the estimated probability of failure using a 95% confidence level. How do I do this, and what assumptions of the distribution should I consider?

I realize that this may be a trivial problem, but I have doubt about some assumptions. Since my work has recently highly relied on these CDF responses to make important decisions, I would like to gain more confidence on how to complete these distributions and provide the resources/references that support these statistical conclusions.


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