Poisson Distribution
Poisson distribution is widely used in quality and reliability engineering. A random variable X has the Poisson distribution with parameter λ, λ>0, the pmf (shown in Figure 1.2) of X is as follows:
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,
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:
The cdf of a continuous distribution is given by
The mean, μ, and variance, σ2, of the continuous random variable are calculated by
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
Figure 1.3 The pdf of the exponential distribution with λ=1.
where λ>0 is the parameter of the distribution. The cdf of the exponential distribution is
If T denotes the failure time of an item with exponential distribution, the reliability function will be
The hazard rate function is
The mean, μ, and variance, σ2 are
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
Figure 1.4 The pdf of the Weibull distribution with β=1.79, η=1.
where β>0 is the shape parameter and η>0 is the scale parameter of the distribution. The cdf of the Weibull distribution is
If T denotes the time to failure of an item with Weibull distribution, the reliability function will be
The hazard rate function is
The mean, μ, and variance, σ2, are
