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The survival function, denoted as \( S(t) \), measures the probability that a given item lasts longer than time \( t \). For a random variable \( X \), the survival function is defined as \( S_X(x) = P(X > x) \). In simpler terms, it tells you how likely it is that the component will still be working after a certain amount of time. If you have a joint probability for two components, you can find their joint survival by multiplying the individual survival functions, assuming they're independent. This is useful to determine the combined likelihood of neither component failing in a given period of time.

The expected value (or mean) of a random variable gives you a measure of the central tendency. It's like an average value you can expect over many trials. For a continuous random variable \( X \) with probability density function (pdf) \( f(x) \), the expected value is calculated as: \[ E(X) = \int_{0}^{\infty} x f(x) \, dx \]In our context, if we have two components with their expected lifetimes \(E(X)\) and \(E(Y)\), you can determine which component is likely to last longer by comparing these values. When solving the given problem, we calculated the expected lifetimes as \( 2 \) years for component \( X \) and \( 5 \) years for component \( Y \). So, on average, component \( Y \) lasts 3 years longer than component \( X \). This concept helps in making informed decisions on warranty periods and reliability assessments.

In probability theory, two variables are independent if the occurrence of one does not affect the occurrence of the other. Mathematically, if \( X \) and \( Y \) are independent, then their joint probability density function \( f_{X,Y}(x, y) \) can be expressed as the product of their individual density functions: \[ f_{X,Y}(x, y) = f_X(x) \, f_Y(y) \]In our case, given the joint pdf \( f(x, y) = 0.1 e^{-x/2} e^{-y/5} \), the independence means that knowing the lifespan of one component provides no information about the lifespan of the other. This property simplifies calculations, such as finding the joint survival function or the probability of failure within a specified period.

Appliance failure probability is about determining the chance that at least one crucial component fails within a specified period. To find this probability, we can use the complement rule. First, calculate the survival probabilities of each component not failing within a given time frame, then find the joint survival probability (if they are independent). For the given components, the probability that both last more than 5 years is: \[ P(X > 5 \text{ and } Y > 5) = S_X(5) * S_Y(5) = e^{-2.5} * e^{-1} = e^{-3.5} \]Using the complement rule: \[ P(X \leq 5 \text{ or } Y \leq 5) = 1 - P(X > 5 \text{ and } Y > 5) = 1 - e^{-3.5} \]This result shows the likelihood of at least one failure occurring within 5 years. Understanding these probabilities is vital for designing reliable products and setting appropriate warranty terms.