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When dealing with multiple random variables, we often need to understand their behavior collectively. Here, the concept of a joint probability density function (PDF) comes into play. The joint PDF describes the likelihood of two or more random variables occurring simultaneously.
Given that the lifetime of two bulbs, represented by random variables \(X\) and \(Y\), are independent and exponentially distributed with a parameter \(\lambda = 1\), the joint PDF can be determined. Each bulb's individual PDF is \(f_X(x) = e^{-x}\) and \(f_Y(y) = e^{-y}\). Since \(X\) and \(Y\) are independent, their joint PDF is the product of the two individual PDFs:

• \(f_{X,Y}(x,y) = f_X(x) \times f_Y(y) = e^{-x} \times e^{-y} = e^{-(x+y)}\)

This joint PDF is valid for \(x \geq 0\) and \(y \geq 0\), capturing how both bulbs will behave together across time. The joint PDF is a foundational step towards making further probability calculations about \(X\) and \(Y\).

Understanding independent random variables is crucial in probability studies. Two random variables \(X\) and \(Y\) are considered independent if the occurrence of one does not affect the occurrence of the other. In simpler terms, knowing the value of \(X\) gives no information about the value of \(Y\), and vice versa.
For our lightbulb problem, the assumption that \(X\) and \(Y\) are independent allows simplifications in calculating joint probabilities. Due to this independence, the joint PDF of \(X\) and \(Y\) is the straightforward product of the marginal PDFs: \(f_{X,Y}(x,y) = e^{-(x+y)}\).
This property simplifies many probability calculations, making it easier to understand and predict the behavior of \(X\) and \(Y\) together. When working with independent variables, integration becomes more manageable since it can be applied separately to each variable.

Calculating probabilities using joint PDFs involves integration over the desired region. For example, if we want to find the probability that each of the two bulbs lasts at most 1000 hours, we calculate \(P(X \leq 1, Y \leq 1)\) by integrating the joint PDF \(f_{X,Y}(x,y)\) over the interval \([0,1]\) for both \(x\) and \(y\).
Using the calculated joint PDF, this results in:

• \(\int_{0}^{1}\int_{0}^{1} e^{-(x+y)}\, dy\, dx\)

This method of evaluating probabilities by integrating the joint PDF over specific boundaries allows us to derive exact probabilities, expanding our understanding of the life expectancy of each bulb.
Other probability calculations, such as the probability that the total lifetime \(X+Y\) is at most 2, similarly involve integrating over the geometric region defined by the condition \(x+y \leq 2\). These calculations give us insights into the combined performance of the bulbs.

Integration plays a pivotal role in determining probabilities from continuous probability density functions, such as those that describe exponential distributions. By integrating the PDF over a certain range, we find the probability that a variable falls within that range.
In problems like the total lifetime of two bulbs \(X+Y\leq 2\), we first identify the region of integration on the \((x, y)\) plane. This involves comprehending the geometric interpretation of conditions, such as \(x+y \leq 2\).
We then perform the double integral of joint PDFs over this region:

• \(\int_{0}^{2} \int_{0}^{2-x} e^{-(x+y)} \, dy \, dx\)

Through evaluating these integrals, we determine the probability that combines events relating to both random variables. Mastering integration in probability solidifies understanding of how regions correspond to probabilities, ensuring accurate and insightful predictions of the variables' behaviors.