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In probability theory, the Joint Probability Density Function (PDF) is a function used to describe the likelihood of two or more random variables occurring simultaneously. For independent random variables, calculating the joint PDF involves multiplying the individual PDFs. In the case of exponential distributions, like the ones described in the problem, each random variable is described by its own exponential PDF.

To find the Joint PDF of two exponential random variables, say \(Y_1\) and \(Y_2\), one simply takes the product of their individual densities. For our problem, each random variable follows \(f(y) = \frac{1}{3}e^{-y/3}\) for \(y > 0\). Hence, the joint PDF becomes:

• \(f_{Y_1,Y_2}(y_1, y_2) = \left(\frac{1}{3}e^{-y_1/3}\right) \left(\frac{1}{3}e^{-y_2/3}\right)\).
• Resulting in \(\frac{1}{9}e^{-(y_1 + y_2)/3}\) for \(y_1, y_2 > 0\).

Understanding joint PDFs is critical, especially when dealing with correlated events in statistics, as they lay the groundwork for calculating probabilities involving multiple random variables.

Random variables are termed "independent" if the occurrence of one provides no information about the occurrence of another. This means that any event related to one variable doesn't affect the likelihood of events related to the other.

In statistical modeling, this assumption simplifies computations significantly. When dealing with independent exponential random variables like in our exercise, their joint distribution directly reflects their individual characteristics without any interaction term. Therefore:

• The joint PDF as a product \(f_{Y_1,Y_2}(y_1, y_2)\) confirms their independence.
• Events involving such variables can be calculated separately and combined simply by multiplication.

This independence is key in applications such as reliability analysis. For example, modeling the life span of two separate fuses highlights their behaviors unaffected by each other. Recognizing independent random variables is essential for building accurate probabilistic models.

When dealing with integration in probability, especially with joint PDFs, properly defining integration limits is crucial. These limits describe the region over which the PDF is integrated to find probabilities.

In our problem, the goal is to find \(P(Y_1 + Y_2 \leq 1)\), which leads to defining a region in the \(y_1\)-\(y_2\) plane to integrate over. Given:

• The constraint \(Y_1 + Y_2 \leq 1\) forms a right-angled triangle with vertices at \((0,0)\), \((1,0)\), and \((0,1)\).
• The integration limits become \(0 < y_1 < 1\) and \(0 < y_2 < 1 - y_1\).

Correctly setting these boundaries ensures the calculation accounts for all relevant probability in the specified region. Visualizing these limits often helps, making it easier to handle complex multi-variable integrals that arise in joint distributions. Understanding integration limits is critical for successful application of probability theory in real-world scenarios.