Reliability Analysis, Safety Assessment and Optimization. Enrico Zio

Reliability Analysis, Safety Assessment and Optimization - Enrico Zio


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      Figure 1.2 The pmf of the Poisson distribution with λ=0.6.

      The mean and variance of the Poisson distribution are

      1.2.2 Continuous Probability Distributions

      We say that a random variable X has a continuous distribution or that X is a continuous random variable if there exists a nonnegative function f, defined on the real line, such that for every interval of real numbers (bounded or unbounded), the probability that X takes a value in an interval [a, b] is the integral of f over that interval, that is,

      upper P left-parenthesis a less-than-or-equal-to upper X less-than-or-equal-to b right-parenthesis equals integral Subscript a Superscript b Baseline f left-parenthesis x right-parenthesis d x period(1.19)

      If X has a continuous distribution, the function f will be the probability density function (pdf) of X. The pdf must satisfy the following requirements:

      f left-parenthesis x right-parenthesis greater-than-or-equal-to 0 comma for all x period(1.20)

      The cdf of a continuous distribution is given by

      The mean, μ, and variance, σ2, of the continuous random variable are calculated by

      sigma squared equals integral Subscript negative infinity Superscript infinity Baseline left-parenthesis x minus mu right-parenthesis squared f left-parenthesis x right-parenthesis d x period(1.22)

      1.2.2.1 Exponential Distribution

      A random variable T follows the exponential distribution if and only if the pdf (shown in Figure 1.3) of T is

      f left-parenthesis t right-parenthesis equals lamda e Superscript minus lamda t Baseline comma t greater-than-or-equal-to 0 comma(1.23)

      where λ>0 is the parameter of the distribution. The cdf of the exponential distribution is

      upper F left-parenthesis t right-parenthesis equals e Superscript minus lamda t Baseline comma t greater-than-or-equal-to 0 period(1.24)

      If T denotes the failure time of an item with exponential distribution, the reliability function will be

      upper R left-parenthesis t right-parenthesis equals 1 minus e Superscript minus lamda t Baseline comma t greater-than-or-equal-to 0 period(1.25)

      The hazard rate function is

      h left-parenthesis t right-parenthesis equals lamda period(1.26)

      The mean, μ, and variance, σ2 are

      sigma squared equals StartFraction 1 Over lamda squared EndFraction period(1.27)

      1.2.2.2 Weibull Distribution

      A random variable T follows the Weibull distribution if and only if the pdf (shown in Figure 1.4) of T is

      f left-parenthesis t right-parenthesis equals StartFraction beta t Superscript beta minus 1 Baseline Over eta Superscript beta Baseline EndFraction e Superscript minus left-parenthesis StartFraction t Over eta EndFraction right-parenthesis Super Superscript beta Superscript Baseline comma t greater-than-or-equal-to 0 comma(1.28)

      where β>0 is the shape parameter and η>0 is the scale parameter of the distribution. The cdf of the Weibull distribution is

      upper F left-parenthesis t right-parenthesis equals 1 minus e Superscript minus left-parenthesis StartFraction t Over eta EndFraction right-parenthesis Super Superscript beta Superscript Baseline comma t greater-than-or-equal-to 0 period(1.29)

      If T denotes the time to failure of an item with Weibull distribution, the reliability function will be

      upper R left-parenthesis t right-parenthesis equals e Superscript minus left-parenthesis StartFraction t Over eta EndFraction right-parenthesis Super Superscript beta Superscript Baseline comma t greater-than-or-equal-to 0 period(1.30)

      The hazard rate function is

      h left-parenthesis t right-parenthesis equals StartFraction beta Over eta EndFraction left-parenthesis StartFraction t Over eta EndFraction right-parenthesis Superscript beta minus 1 Baseline comma t greater-than-or-equal-to 0 period(1.31)

      The mean, μ, and variance, σ2, are

      sigma squared equals eta squared left-bracket upper Gamma left-parenthesis StartFraction 2 plus beta Over beta EndFraction right-parenthesis minus left-parenthesis upper Gamma left-parenthesis StartFraction 1 plus beta Over beta EndFraction right-parenthesis right-parenthesis squared right-bracket period(1.32)

      1.2.2.3 Gamma Distribution

      A random variable T follows the gamma distribution if and only if the pdf (shown in Figure 1.5) of T is

      f left-parenthesis t right-parenthesis equals StartFraction lamda Superscript beta Baseline Over upper Gamma left-parenthesis beta right-parenthesis EndFraction t Superscript beta minus 1 Baseline e Superscript minus lamda t Baseline comma t greater-than-or-equal-to 0 comma(1.33)

      where β>0 is the shape parameter and η>0 is the scale parameter of the distribution. The cdf of the gamma distribution is

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