Applied Probability and Stochastic Processes by Bryc

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We may want to assume that a component analyzed becomes less reliable, or more reliable with time. An example of the rst one is perhaps a brand new car. An example of the latter is perhaps a software operating system when updates are installed promptly. Let T 0 be a continuous r. v. interpreted as a failure time of a certain device. If the device is in operational condition at time t, then the probability that it will fail immediately afterwards may be assumed negligible. The probability of failing within h units of time is PrT t + hjT t.

The probability of failing within h units of time is PrT t + hjT t. 31 If T is exponential then the failure rate is constant. A family of failure rates that exhibit interesting aging patterns is provided by the family of power functions t = ta . 1 If T is continuous with failure rate t = ta , where a 0 then T has the Weibull density: f t = Cta,1 e,bta for t 0: Here C = ab is the normalization constant. 17 Poisson approximation Of the discrete distributions, the formula for the Poisson distribution is perhaps mysterious.

As an example, suppose that we would like to model the birthday problem with people born non-uniformly throughout the year. Which way would you expect the answer to change? 2. What are typical errors of a simulation of size n? How can we estimate the accuracy of the answer without having the exact answer to compare it to? Chapter 5 gives theoretical basis for such estimates. 19 Exercise A family has two children, and one of them is a boy. What is the probability that they have two boys? 20 Exercise A die is thrown until an ace turns up.

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