can be rewritten as
In all generality, the expected value or mean of the time to failure T is called the mean time to failure (MTTF), which is defined as
It is equivalent to
Another related concept is the mean time between failures (MTBF). MTBF is the average working time between two consecutive failures. The difference between MTBF and MTTF is that the former is used only in reference to a repairable item, while the latter is used for non-repairable items. However, MTBF is commonly used for both repairable and non-repairable items in practice.
The failure rate function or hazard rate function, denoted by h(t), is defined as the conditional probability of failure in the time interval [t, t+Δt] given that it has been working properly up to time t, which is given by
Furthermore, the cumulative failure rate function, or cumulative hazard function, denoted by H(t), is given by
Example 1.2 The failure time of a valve follows the exponential distribution with parameter h(t) (in arbitrary units of time-1). The value is new and functioning at time h(t). Calculate the reliability of the valve at time h(t) (in arbitrary units of time).
Solution
The pdf of the failure time of the valve is
The reliability function of the valve is given by
At time, the value of the reliability is
1.2 Component Reliability Modeling
As mentioned in the previous section, in reliability engineering, the time to failure of an item is a random variable. In this section, we briefly introduce several commonly used discrete and continuous distributions for component reliability modeling.
1.2.1 Discrete Probability Distributions
If random variable X can take only a finite number k of different values x1,x2,…,xk or an infinite sequence of different values x1,x2,…, the random variable X has a discrete probability distribution. The probability mass function (pmf) of X is defined as the function f such that for every real number x,
If x is not one of the possible values of X, then f(x)=0. If the sequence x1,x2,… includes all the possible values of X, then ∑if(xi)=1. The cdf is given by
1.2.1.1 Binomial Distribution
Consider a machine that produces a defective item with probability p (0<p<1) and produces a non-defective item with probability 1−p. Assume the events of defects in different items are mutually independent. Suppose the experiment consists of examining a sample of n of these items. Let X denote the number of defective items in the sample. Then, the random variable X follows a binomial distribution with parameters n and p and has the discrete distribution represented by the pmf in (1.14), shown in Figure 1.1. The random variable with this distribution is said to be a binomial random variable, with parameters n and p,
Figure 1.1 The pmf of the binomial distribution with n=5, p=0.4.
The pmf of the binomial distribution is
For a binomial distribution, the mean, μ, is given by
and the variance, σ2, is given by
1.2.1.2
