Template:Distribution ex rsa

Distribution Example
Visualize a socket into which a component is inserted at time $$0$$. When the component fails, it is replaced immediately with a new one of the same kind. After each failure, the socket is put back into an as good as new condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a renewal process. In the illustration below, the component life is $${{X}_{j}}$$  and  $${{t}_{j}}$$  is the system time to the  $${{j}^{th}}$$  failure. Each component life $${{X}_{j}}$$  in the socket is governed by the same distribution  $$F(x)$$. A distribution, such as the Weibull, governs a single lifetime. There is only one event associated with a distribution. The distribution $$F(x)$$  is the probability that the life of the component in the socket is less than  $$x$$. In the illustration above, $${{X}_{1}}$$  is the life of the first component in the socket. $$F(x)$$ is the probability that the first component in the socket fails in time  $$x$$. When the first component fails, it is replaced in the socket with a new component of the same type. The probability that the life of the second component is less than $$x$$  is given by the same distribution function,  $$F(x)$$. For the Weibull distribution:


 * $$F(x)=1-{{e}^{-\lambda {{x}^{\beta }}}}$$

A distribution is also characterized by its density function, such that:


 * $$f(x)=\frac{d}{dx}F(x)$$

The density function for the Weibull distribution is:


 * $$f(x)=\lambda \beta {{x}^{\beta -1}}\cdot {{e}^{-\lambda \beta x}}$$

In addition, an important reliability property of a distribution function is the failure rate, which is given by:


 * $$r(x)=\frac{f(x)}{1-F(x)}$$

The interpretation of the failure rate is that for a small interval of time $$\Delta x$$,  $$r(x)\Delta x$$  is approximately the probability that a component in the socket will fail between time  $$x$$  and time  $$x+\Delta x$$ , given that the component has not failed by time  $$x$$. For the Weibull distribution, the failure rate is given by:


 * $$r(x)=\lambda \beta {{x}^{\beta -1}}$$

It is important to note the condition that the component has not failed by time $$x$$. Again, a distribution deals with one lifetime of a component and does not allow for more than one failure. The socket has many failures and each failure time is individually governed by the same distribution. In other words, the failure times are independent of each other. If the failure rate is increasing, then this is indicative of component wearout. If the failure rate is decreasing, then this is indicative of infant mortality. If the failure rate is constant, then the component failures follow an exponential distribution. For the Weibull distribution, the failure rate is increasing for $$\beta >1$$, decreasing for  $$\beta $$   $$<1$$  and constant for  $$\beta =1$$. Each time a component in the socket is replaced, the failure rate of the new component converts back to the value at time $$0$$. This means that the socket is as good as new after each failure and the subsequent replacement by a new component. This process is continued for the operation of the socket.