The joint density function for a pair of random variables X and Y is given. (Round your answers to four decimal places.)
f(x, y) =

Cx(1 + y)	    if 0 ≤ x ≤ 4, 0 ≤ y ≤ 2
0	    otherwise

To find the value of \( C \) that makes \( f(x, y) \) a valid joint density function, we need to ensure that the total probability over the entire range is 1. This means we need to integrate \( f(x, y) \) over the given range and set the result equal to 1.

The joint density function is given by:
\[ f(x, y) = \begin{cases} 
Cx(1 + y) & \text{if } 0 \leq x \leq 4, \, 0 \leq y \leq 2 \\
0 & \text{otherwise} 
\end{cases} \]

To find \( C \), we integrate \( f(x, y) \) over the specified range:
\[ \int_{0}^{2} \int_{0}^{4} Cx(1 + y) \, dx \, dy = 1 \]

First, integrate with respect to \( x \):
\[ \int_{0}^{4} Cx(1 + y) \, dx = C(1 + y) \int_{0}^{4} x \, dx \]
\[ = C(1 + y) \left[ \frac{x^2}{2} \right]_{0}^{4} \]
\[ = C(1 + y) \left( \frac{4^2}{2} - \frac{0^2}{2} \right) \]
\[ = C(1 + y) \left( \frac{16}{2} \right) \]
\[ = C(1 + y) \cdot 8 \]

Now, integrate with respect to \( y \):
\[ \int_{0}^{2} 8C(1 + y) \, dy \]
\[ = 8C \int_{0}^{2} (1 + y) \, dy \]
\[ = 8C \left[ y + \frac{y^2}{2} \right]_{0}^{2} \]
\[ = 8C \left( 2 + \frac{2^2}{2} - (0 + \frac{0^2}{2}) \right) \]
\[ = 8C \left( 2 + \frac{4}{2} \right) \]
\[ = 8C \left( 2 + 2 \right) \]
\[ = 8C \cdot 4 \]
\[ = 32C \]

Set this equal to 1:
\[ 32C = 1 \]
\[ C = \frac{1}{32} \]

Thus, the value of \( C \) is \( \frac{1}{32} \).

### Summary
The value of \( C \) that makes \( f(x, y) \) a valid joint density function is:
\[ C = \frac{1}{32} \]

Would you like more details or have any questions?

Here are 5 numbered questions you might want to ask:

1. How do you verify if a given joint density function is valid?
2. Can you show a step-by-step example of integrating a joint density function?
3. What are marginal density functions, and how do you find them from a joint density function?
4. How do you find the expected value of a function of two random variables?
5. What is the significance of the joint density function in probability theory?

**Tip:** When working with joint density functions, always ensure that the total probability over the defined range is 1, as this validates the function as a proper density function.

Learn how to find the value of C that makes the joint density function f(x, y) valid, with a step-by-step solution. This problem covers the concepts of joint density functions and integration, ensuring the total probability over the range is 1.

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