Question

Consider \(n\) components with independent lifetimes, which are such that component \(i\) functions for an exponential time with rate \(\lambda_{i} .\) Suppose that all components are initially in use and remain so until they fail. (a) Find the probability that component 1 is the second component to fail. (b) Find the expected time of the second failure. Hint: Do not make use of part (a).

Short answer

(a) The probability that component 1 is the second component to fail is given by the following formula: \[P(\text{component 1 fails second}) = \sum_{i \neq 1}^{n} \int_0^\infty \int_0^{t_2} P_{i,1}(t_1, t_2) dt_1 dt_2,\] where \(P_{i,1}(t_1, t_2)\) is calculated using the probability density function of each component and the first-failure probabilities. (b) The expected time of the second failure is given by: \[E[T_2] = \int_0^\infty t_2 \, h(t_2) dt_2,\] where \(h(t_2)\) is the probability density function (PDF) of the time until the second failure, found by computing the marginal distribution of the joint PDF \(g(t_1, t_2)\).

Step-by-step solution

Compute the probability of a component failure rate

To find the probability that component 1 is the second component to fail, we need to first compute the probability of a component failing at a specific rate. Given that component \(i\) functions for an exponential time with rate \(\lambda_i\), its probability density function (PDF) is: \[f_i(t) = \lambda_i e^{-\lambda_i t},\] for \(t \ge 0\).

Compute the probability of any one component failing first

Now, we need to compute the probability that any one component fails first. Since there are \(n\) independent components, the probability that component \(i\) fails first is given by the product of the probability that component \(i\) fails and the probability that all other components survive until that time. Let \(T_i\) represent the time to the first failure in the absence of component \(i\). \[P_i(t) = f_i(t) \prod_{j \neq i}^{n} (1 - F_j(t))\] where \(F_j(t)\) is the cumulative distribution function (CDF) of component \(j\), given by: \[F_j(t) = 1 - e^{-\lambda_j t}\]

Compute the probability that component 1 is the second component to fail

To compute this probability, we first need to find the probability of any two components failing in a specific order. To do this, we can use the probability density function of each component and the previously calculated first-failure probability. \[P_{1,2}(t_1, t_2) = P_1(t_1) \times f_2(t_2 - t_1) \prod_{j \neq 1,2}^{n} (1 - F_j(t_2))\] Since we are interested in the probability that component 1 is the second component to fail, we can sum the probabilities of all possible orders that result in component 1 failing second: \[P(\text{component 1 fails second}) = \sum_{i \neq 1}^{n} \int_0^\infty \int_0^{t_2} P_{i,1}(t_1, t_2) dt_1 dt_2\] (b) Finding the expected time of the second failure

Compute the joint PDF of the time until the first two failures

To find the expected time until the second failure, we need to compute the joint probability density function (PDF) of the time until the first and second failures: \[g(t_1, t_2) = \sum_{1 \le i < j \le n} P_{i, j}(t_1, t_2),\] with \(0 \le t_1 \le t_2\).

Compute the PDF of the time until the second failure

Now, we need to compute the probability density function (PDF) of the time until the second failure. This is given by the marginal distribution of the joint PDF \(g(t_1, t_2)\): \[h(t_2) = \int_0^{t_2} g(t_1, t_2) dt_1\]

Compute the expected time of the second failure

After obtaining the distribution of the second failure time, we can calculate its expected value: \[\text{Expected time of second failure} = E[T_2] = \int_0^\infty t_2 \, h(t_2) dt_2\] This integral will give us the expected time for the second failure.

Key concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in statistics and is particularly important in understanding the exponential distribution. It depicts the likelihood of a continuous random variable to assume a specific value. For the exponential distribution, the PDF is expressed mathematically as \(f_i(t) = \lambda_i e^{-\lambda_i t}\), where \(\lambda_i\) is the rate parameter, and \(t\) is time.

This function captures how likely it is for a component to fail at each moment in time, given that it has survived up to that moment. To actually use the PDF to find probabilities, we need to integrate over a given range of \(t\). However, for discrete events such as 'the second component to fail', we often need more than just the PDF - we need to combine it with the failure rates of other components and other concepts, like the cumulative distribution function and expected value.

It's crucial to remember that the area under the PDF curve across all possible values of \(t\) (from 0 to infinity) equals 1. This is because a component will eventually fail, making the probability of a failure event 100% over an infinite timeline.

Cumulative Distribution Function

While the probability density function helps identify the probability of failure at a specific time, the cumulative distribution function (CDF) takes a broader view by answering this question: What is the probability that the component has failed by time \(t\)? This is crucial for understanding the timing of the first failure and subsequent failures in a set of components.

The CDF, denoted as \(F_j(t)\), is derived from the PDF and for an exponential distribution is written as \(F_j(t) = 1 - e^{-\lambda_j t}\). It starts at 0 when \(t = 0\) and asymptotically approaches 1 as \(t\) goes to infinity, signifying certainty of failure over time. Understanding the CDF is essential for solving problems involving the order of failures among multiple components, such as determining the likelihood of which component fails first, second, and so on.

In practice, we use the CDF to calculate probabilities over intervals. For instance, the probability that a component lasts longer than a certain time but fails before another specified time can be found by subtracting the CDF values at these two points.

Expected Value

The expected value is a concept that allows us to predict the average outcome of a random process in the long run. For the exponential distribution, the expected value is particularly straightforward—it is simply the reciprocal of the rate parameter, \(1/\lambda_i\), which gives the average time until failure for a single component.

However, when dealing with multiple components, things can become more complicated, as we must account for the different rates at which each component may fail. In the exercise, the expected time of the second failure is not just the average of individual expected values; instead, it requires the integration of a complex probability density function that considers the interactions between components' lifetimes and failure rates.

Mathematically, to find the expected time of the second failure, you compute the weighted average of all possible second failure times, using the probability of each second failure time occurring as weights. This is represented by the integral \(E[T_2] = \int_0^\infty t_2 \, h(t_2) dt_2\), where \(h(t_2)\) is the PDF specifically for the time of the second failure. Understanding this concept is paramount for practical applications like reliability engineering and risk assessment.