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When confronted with a problem involving double integration, it's important to start with the basics: understanding what it means. Double integration is the process of integrating a function of two variables, say, for example, \(x\) and \(y\), over a two-dimensional region. This is akin to finding the 'total weight' if you can imagine the function as representing a density over a surface area.

In the context of joint probability density functions, double integration allows us to calculate the probability of a certain event happening within a defined region. The area under the curve (or surface, in this case) between specific bounds gives us the probability we are seeking. Conceptually, imagine you are painting a specific section of a wall; the amount of paint used over the area can be viewed as the 'probability weight.', and double integration is akin to calculating the amount of paint needed based on the density of the paint coverage. This is why in our exercise the computation of the double integral of the given joint pdf \(f_{X,Y}(x,y)\) over the specified region yields the probability in question.

The concept of probability in mathematical terms is a measure of how likely an event is to occur. It ranges from \(0\) (impossible event) to \(1\) (certain event). The joint probability density function (pdf) we are dealing with in the context of the given exercise describes the likelihood of two random variables, \(X\) and \(Y\), taking on certain values within their respective ranges.

To determine the probability \(P(Y<3X)\), we integrate the joint pdf over the range where this condition is true. Understanding that integration over the joint pdf gives us the probability of the variables falling within a certain range is essential. It's similar to asking, 'What are the chances it will rain today given the current weather patterns?' but instead, you are asking, 'What is the likelihood these variables fall into this specific range under the given conditions?'. It is this integration that quantifies the likelihood (or probability) of our event of interest.

The bounds of integration set the limits within which we are interested in finding the probability. These bounds can be influenced by the conditions in the problem statement. In our exercise, the condition \(P(Y<3X)\) implies that the value of \(Y\) must be less than three times the value of \(X\). Consequently, this defines a triangular region in the first quadrant of the XY-plane, with \(X\) and \(Y\) as positive values.

Carefully determining the correct integration bounds is vital. The bounds for \(x\) go from \(y/3\) to \(y\), reflecting the fact that \(x\) cannot be greater than \(y\), due to the condition \(0