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The concept of 'system lifetime' refers to how long a system can operate before it fails. In this context, we have a system made up of three components. Each component has a certain lifetime, denoted by the random variables \(X_1, X_2,\) and \(X_3\) for components 1, 2, and 3, respectively. The system will continue to function as long as the first component functions, along with at least one of the second or third components. This means the system can still work even if one of the latter two components fails.

Understanding system lifetime requires considering how these components work together. If component 1 stops working, the whole system fails. Alternatively, if both component 2 and 3 fail, the system stops, even if component 1 is still functional. Hence, the system's lifetime is influenced by the combination of these conditions.

To analyze the system lifetime, we need to use probability theory to determine the likelihood the system will run up to a certain time \(y\). This involves formulating the probability that describes how long \(Y\), the system lifetime, will exceed \(y\). This probability uses the properties of independent and exponential distributions in the analysis.

The cumulative distribution function (CDF) is a crucial concept when analyzing system lifetime. The CDF, denoted as \(F(y)\), provides the probability that a random variable, like system lifetime \(Y\), takes on a value less than or equal to \(y\). In our exercise, we are interested in finding \(F(y) = P(Y \leq y)\), which represents the probability that the system fails by time \(y\).

In practical terms, you calculate the CDF by first finding the probability that the system is still operational at time \(y\), denoted as \(P(Y > y)\). From the properties of probability, \(F(y) = 1 - P(Y > y)\). This step often involves events like intersections or unions of component lifetimes, technique used in the original exercise to navigate through complex dependencies.

• Step 1: Identify states where the system is still working. It's the complement of failure, dealing with component conditions.
• Step 2: Translate states into probabilities using exponential distribution. Since \(P(X_i > y) = e^{-\lambda y}\), find each component’s contribution to system operation.
• Step 3: Use these probabilities to determine \(P(Y > y)\) and subsequently \(F(y)\).

By addressing \(F(y)\), you start to capture a complete picture of how the system lifetime behaves under given component conditions.

The exercise involves the concept of independent random variables, which is fundamental when dealing with multiple components in a system. Each component's lifetime is represented by a random variable \(X_i\). Independence means that the behavior or failure of one component does not influence the others.

In mathematical terms, the joint probability of independent events equals the product of their individual probabilities. Thus, if \(X_1\), \(X_2\), and \(X_3\) are independent, we can write: \[ P(X_1 \leq y, X_2 \leq y, X_3 \leq y) = P(X_1 \leq y) \times P(X_2 \leq y) \times P(X_3 \leq y) \]
This property simplifies a lot of probability calculations, making it easier to combine different conditions in system lifetime analysis.

Independence is crucial for calculating system lifetimes because it allows the use of simple multiplication to combine probabilities of different components without worrying about how one component's state affects others. This makes exponential distribution properties handy since the exponential is a memoryless distribution, aligning perfectly with the independence assumption and making analytical computations more straightforward.