By Jingshan Li, Semyon M. Meerkov

ISBN-10: 0387755780

ISBN-13: 9780387755786

ISBN-10: 0387755799

ISBN-13: 9780387755793

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FX (x) Several useful examples of continuous random variables are given below. Exponential random variable. 12, which is a continuous time analogue of the geometric machine. Namely, if it is up (respectively, down) at time t, it goes down (respectively, up) during an infinitesimal time δt with probability λδt (respectively, µδt). The numbers λ and µ are called the breakdown and repair rates, respectively. 2. 12: Exponential machine therefore, can take any positive value. Note, however, that λδt and µδt are probabilities and, therefore must be between 0 and 1.

19) 28 CHAPTER 2. 11: Probability calculations for a continuous random variable The conditional cdf and pdf are defined by FX (x|A) = P [X ≤ x|A] = fX (x|A) = dFX (x|A) , dx P [{X ≤ x} ∩ {A}] , P [A] where A is an event, serving as a condition. 21) i=1 n fX (x) = i=1 where Ai , i = 1, . . , n, is the partitioning. Finally, Bayes’s formulas for cdf and pdf take the form: P [A|x1 < X ≤ x2 ] = P [A|X = x] = [FX (x2 |A) − FX (x1 |A)]P [A] , P [x1 < X ≤ x2 ] fX (x|A)P [A] . fX (x) Several useful examples of continuous random variables are given below.

Another example of the Bernoulli random variable arises in machines, which can produce either good or defective parts. Indeed, assume that P [{part produced by a machine during a cycle time is good}] = g, P [{part produced by a machine during a cycle time is defective}] = 1 − g. If the quality of parts produced in each cycle is independent from that in other cycles, such a machine is referred to as obeying the Bernoulli quality model. This model is appropriate, for example, for automotive painting operations where the defects are due to independent random events, such as dust, scratches, etc.

### Production Systems Engineering by Jingshan Li, Semyon M. Meerkov

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