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Time to failure Probability, Survival, the Hazard rate, and the Conditional Failure Probability

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**f(t) is the Probability Density Function (PDF).**

Failures do not occur at fixed times. They occur randomly according to a probability distribution. The PDF is the usual way of representing a failure distribution (also known as an “age-reliability relationship”). Working Age t

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**f(t) is the Probability Density Function (PDF).**

As density equals mass per unit of volume, probability density is the probability of failure per unit of time. When multiplied by the length of a small time interval T at t, the product is the probability of failure in that interval. The PDF is the basic description of the time to failure of an item. T x f(t) T Working Age t

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**f(t) is the Probability Density Function (PDF).**

The PDF is often estimated from real life data. It resembles a histogram of the failures of an item in consecutive age intervals. All other functions related to an item’s reliability can be derived from the PDF. For example, The area Σ(t x f(t)) under the PDF curve between time 0 and time t1 is the (cumulative) probability F(t) of failing prior to t1. T T T T T t1 Working Age t

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**Finding the PDF equation**

The easiest (standard) way to get the PDF equation is through Weibull Analysis. Weibull Analysis assumes that the equation has the form: is the shape factor, is the scale factor Given a sample of life cycles we can estimate and using numerical methods.

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**All other functions related to an item’s reliability can be derived from the PDF. For example:**

F(t) is the cumulative distribution function (CDF). It is the area under the f(t) curve from 0 to t.. (Sometimes called the unreliability, or the cumulative probability of failure.) R(t) is the survival function. (Also called the reliability function.) R(t) = 1-F(t) h(t) is the hazard rate. (At various times called the hazard function or failure rate.) h(t) = f(t)/R(t) H(t) is the conditional probability of failure H(t)= (R(t)-R(t+L))/R(t). It is the probability that the item fails in a time interval [t to t+L] given that it has not failed up to time t. Its graph resembles the shape of the hazard rate curve. MTTF is the average time to failure. (Also called the mean time to failure, expected time to failure, or average life.)

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**Conditional Failure Probability versus Failure Rate**

Often, the two terms "conditional probability of failure" and "hazard rate" are used interchangeably in many RCM and practical maintenance references. They sometimes define both terms as: “The probability that an item will fail during an age interval given that the item enters (or survives) to that age interval.“ This definition is not the one usually meant in reliability theoretical works when they refer to “hazard rate” or “hazard function”. Nowlan and Heap point out that the hazard rate may be considered as the limit of the ratio (R(t)-R(t+L))/(R(t)*L) as the age interval L tends to zero. This is shown on the next slides.

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To summarize, "hazard rate" and "conditional probability of failure" are often used interchangeably (in more practical maintenance books). The “hazard rate” is commonly used in most reliability theory books. The conditional probability of failure is more popular with reliability practitioners and is used in RCM books such as those of N&H and Moubray. The two definitions are called: "hazard rate“, and "conditional probability of failure“.

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**H(t) = (R(t)-R(t+L))/R(t).**

The two definitions h(t) = f(t)/R(t) H(t) = (R(t)-R(t+L))/R(t). where L is the length of an age interval.

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**H(t) = (R(t)-R(t+L))/R(t).**

h(t) = f(t)/R(t) H(t) = (R(t)-R(t+L))/R(t). When you divide equation 2 by L and let L tend to 0, you get equation 1, as follows:

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**h(t) = f(t)/R(t) (Hazard, Failure rate func.)**

H(t) = (R(t)-R(t+L))/R(t). (Conditional failure prob.) F(t)=1-R(t) (Cumulative failure prob.) Differentiating F(t)=1-R(t) f(t)= -dR(t)/d(t) Dividing the right side of H(t) = (R(t)-R(t+L))/R(t) by L and and then applying the definition of a limit as L tends to 0. Lim R(t)-R(t+L) = (1/R(t))( -dR(t)/dt) = f(t)/R(t) = h(t) L0 LR(t)

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**This applies not only to hazard**

Actually, not only the hazard function, but f(t), F(t), and R(t) also have two versions of their defintions as above. The first version is defined over a continous range of age t while the second one is defined over discrete age intervals, e.g., (0,100), (100,200), (200,300), ... Roughly, we can say the second definition is a discrete version of the first definition.

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**How are h(t) and H(t) used?**

h(t) = f(t)/R(t) is useful in reliability theory and is mainly used for theoretical development. H(t) = (R(t)-R(t+L))/R(t) is useful for reliability practitioners, since in practice people usually divide the age horizon into a number of equal age intervals. The PDF, cdf, and Survival function may all be calculated using age intervals. The results would be similar to histograms, rather than continous functions obtained using the failure rate defintion.

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