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The concept of the Joint Probability Density Function (PDF) is key in probability when dealing with more than one random variable. In the case of Annie and Alvie, who arrive at their meeting place at different times, we have two random variables: - Annie's arrival time, denoted by the random variable \(X\)- Alvie's arrival time, denoted by the random variable \(Y\)If these variables are independent, as is the case here, the joint PDF helps us calculate the likelihood of both events occurring together at specific times. The joint PDF, denoted \(f_{X,Y}(x,y)\), is simply the product of the two individual PDFs when variables are independent.For Annie and Alvie's arrival, we have:- \(f_X(x) = 3x^2\) for \(0 \leq x \leq 1\)- \(f_Y(y) = 2y\) for \(0 \leq y \leq 1\)Thus, their joint PDF is given by:\[f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) = 6x^2y\]This equation holds only within the defined range \(0 \leq x, y \leq 1\). Understanding the joint PDF allows us to explore the combined behavior of the two independent variables.

Integration plays a vital role in probability, especially when calculating expected values such as the expected waiting time for Annie or Alvie. When we want to find the expected value of a function of two continuous random variables, like \(|X-Y|\), integration helps us.To calculate the expected waiting time \(E[|X-Y|]\), we use a double integral over the range of possible values that \(X\) and \(Y\) can take. The expression is:\[E[|X-Y|] = \int_0^1 \int_0^1 |x-y| \cdot 6x^2y \, dy \, dx\]Here, the integrals go from 0 to 1, corresponding to the time between noon and 1:00 PM. The function \(|x-y|\) measures the absolute time difference in their arrivals.The tricky part is considering the regions where \(x > y\) and \(x < y\). Due to symmetry, we only need to compute for \(x > y\) and then double the result. This involves setting up separate integrals and then simplifying them, often requiring substitution and partial fraction techniques.

The concept of independent random variables is crucial in probability. Two random variables are independent if the occurrence of one event does not affect the occurrence of another. For Annie and Alvie's arrival times:- Independence implies that Annie's time \(X\) does not affect Alvie's time \(Y\) and vice versa.Mathematically, this means that their joint PDF is the product of their respective PDFs, \(f_X(x)\) and \(f_Y(y)\). This simplifies calculations immensely:\[f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)\]In our case, Annie's arrival time is modeled by \(3x^2\) and Alvie's by \(2y\).Understanding independence allows us to create models where more complex interactions would otherwise be too complicated or impractical to evaluate. Independence can simplify the distribution and expected value calculations, making them feasible for integration. As with Annie and Alvie, it can help solve real-world problems like determining the expected waiting time without overly complicated math.