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In probability theory, a probability density function (PDF) describes the likelihood of a continuous random variable falling within a particular range of values. The PDF is a fundamental concept because it explains how probabilities are distributed across possible values of a random variable.
In the given exercise, the joint density function is expressed as \[ f(y_1, y_2) = \frac{1}{8} y_1 e^{-(y_1 + y_2)/2}, \quad y_1 > 0,\, y_2 > 0 \] This expression tells us how two continuous random variables, \( Y_1 \) and \( Y_2 \) (representing the lifespan of two types of components in hundreds of hours), behave together. When we say 'joint density,' we're referring to the combined behavior of both variables.

• The joint PDF should be non-negative over its domain.
• The total probability over the domain must equal one for it to be a valid PDF.

Understanding the behavior of these variables through the joint PDF allows us to draw conclusions about the likelihood of component lifespan combinations in the system.

Integration is a mathematical process used to calculate totals, areas, or volumes. In the context of probability density functions, integration helps to determine probabilities. When dealing with a continuous random variable, integration is used to compute the area under the curve of the PDF.
### Computing Total ProbabilityThe integration performed in the exercise ensures that the total probability is equal to one, thus confirming the function's validity.
1. **Integrate with Respect to \( y_2 \):** This step involves integrating the function \[ \frac{1}{8} y_1 e^{-(y_1+y_2)/2} \] from \( y_2 = 0 \) to \( y_2 = \infty \). The result here is \[ \int_0^\infty e^{-y_2/2} \, dy_2 = 2 \].
2. **Integrate with Respect to \( y_1 \):** After obtaining the result of the first integration, the remaining expression \[ \frac{1}{4} y_1 e^{-y_1/2} \] is integrated over \( y_1 \) from 0 to \( \infty \), yielding:\[ \int_0^\infty y_1 e^{-y_1/2} \, dy_1 = 4 \].

This integration steps demonstrate that the integral of the joint PDF across its entire domain is equal to one.

In probability and statistics, a random variable is a variable that can take on different values, each with a certain probability. Random variables are fundamental to understanding statistical experiments and real-world probability scenarios.
### Types of Random Variables

• **Discrete Random Variables:** These take on distinct, separate values, often as counts or listed values.
• **Continuous Random Variables:** These can take any value within a range, which is often represented by intervals on a number line.

In the exercise, \( Y_1 \) and \( Y_2 \) are examples of continuous random variables.
**Joint Random Variables:** When two or more random variables are considered together, they become joint random variables. Joint distributions illustrate the behavior and relationships between multiple random variables.
In this context, \( Y_1 \) and \( Y_2 \) together form a joint random distribution which helps to understand how the lifetimes of components I and II vary in relation to each other.