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When faced with a joint probability density function (PDF), one key goal is to integrate it over specified regions to find probabilities. Joint PDFs describe the probability distribution of two random variables happening simultaneously. In our example, we have a function defined as \( f(x, y) = Cx(1+y) \) over the area where \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 2 \). To analyze this further, we must integrate over the specified areas to understand the behavior of the probability distribution.Consider the full region for the variables \( X \) and \( Y \), which is between 0 and 1 for \( x \), and 0 and 2 for \( y \):

• First, perform the integral with respect to \( x \), keeping \( y \) constant.
• Next, integrate the resulting expression with respect to \( y \).

The objective here is to calculate the overall probability which should total to 1 in the given range. These steps allow us to see where the function holds true and outline the region where \( X \) and \( Y \) can most likely occur in tandem. This is vital for finding the normalization constant \( C \).

To ensure a probability density function is valid, its integral over the complete space must equal 1. This requires us to find the "constant" that adjusts the function to suit this rule. In our example:\[\int_0^2\int_0^1 Cx(1+y) \, dx \, dy = 1\]

Steps to Find the Constant:

• Integrate \( Cx(1+y) \) with respect to \( x \), setting the bounds from 0 to 1, while treating \( y \) as constant. This usually involves basic integral calculus formulas.
• The resulting expression will then be integrated with respect to \( y \), from 0 to 2.
• Set the completed integral equal to 1, since the total probability across the defined space should be complete (one).

Applying these steps reveals \( C = \frac{1}{2} \), which normalizes the joint PDF in the defined range. Consider this as rebalancing weights to ensure uniform application across possibilities within the range.

Once the joint PDF is normalized, calculating specific probabilities involves integrating over particular regions of interest.For example, to find \( P(X \leq 1, Y \leq 1) \), integrate the function over the area where \( x \) and \( y \) both lie between 0 and 1:

• The integral's bounds for both \( x \) and \( y \) are the same, simplifying the region to a square.

Resulting in a probability of \( \frac{3}{8} \).Another interesting region represents \( P(X+Y \leq 1) \). This inequality describes a triangular area on the \( xy \)-plane:

• For this, the bounds of \( x \) are \( 0 \) to \( 1-y \), with \( y \) running from \( 0 \) to \( 1 \).

Calculating this, you find \( \frac{59}{96} \) as the probability. By understanding and applying these step-by-step integrations over particular ranges, determining the probabilities for complex sets of conditions becomes manageable.