Exercise 2.9. Suppose (X, Y ) is uniform over the rectangle [0, 2] × [0, 5]. What is the
average of x2y over this interval? Hint: shorten your work a little with the magic product
formula.

We are asked to find the average of \( x^2 y \) over the rectangular region \([0, 2] \times [0, 5]\), where \((X, Y)\) is uniformly distributed. The average or expected value of \( x^2 y \) for a uniform distribution is calculated as:

\[
\mathbb{E}[X^2 Y] = \frac{1}{\text{Area}} \int_0^2 \int_0^5 x^2 y \, dy \, dx
\]

### Step 1: Determine the area of the rectangle
The area of the rectangle is:

\[
\text{Area} = 2 \times 5 = 10
\]

### Step 2: Set up the double integral
We need to compute the double integral:

\[
\mathbb{E}[X^2 Y] = \frac{1}{10} \int_0^2 \int_0^5 x^2 y \, dy \, dx
\]

### Step 3: Evaluate the inner integral with respect to \( y \)
First, we integrate \( x^2 y \) with respect to \( y \) from 0 to 5:

\[
\int_0^5 x^2 y \, dy = x^2 \left[ \frac{y^2}{2} \right]_0^5 = x^2 \left( \frac{5^2}{2} - \frac{0^2}{2} \right) = x^2 \cdot \frac{25}{2} = \frac{25}{2} x^2
\]

### Step 4: Evaluate the outer integral with respect to \( x \)
Now, we integrate the result with respect to \( x \) from 0 to 2:

\[
\int_0^2 \frac{25}{2} x^2 \, dx = \frac{25}{2} \int_0^2 x^2 \, dx = \frac{25}{2} \left[ \frac{x^3}{3} \right]_0^2 = \frac{25}{2} \cdot \frac{8}{3} = \frac{25 \times 8}{6} = \frac{200}{6} = \frac{100}{3}
\]

### Step 5: Divide by the area to find the average
Finally, divide by the area of the rectangle (which is 10):

\[
\mathbb{E}[X^2 Y] = \frac{1}{10} \times \frac{100}{3} = \frac{100}{30} = \frac{10}{3}
\]

### Final Answer:
The average of \( x^2 y \) over the interval \([0, 2] \times [0, 5]\) is \( \frac{10}{3} \).

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Would you like more details or clarifications on any part of the solution?

Here are 5 related questions for further practice:
1. What is the expected value of \(X^2\) over the interval [0, 2]?
2. What is the expected value of \(Y\) over the interval [0, 5]?
3. How would the result change if the region were [0, 3] × [0, 4]?
4. Can you compute the average of \(X^3 Y^2\) over the same region?
5. What is the expected value of \(XY\) over the rectangle [0, 2] × [0, 5]?

**Tip:** For double integrals, it's useful to evaluate the inner integral first before working on the outer one to simplify the overall computation.

Suppose (X, Y) is uniform over the rectangle [0, 2] × [0, 5]. What is the average of x^2y over this interval? Hint: shorten your work a little with the magic product formula.

Explore the calculation of the average of x^2y over the uniform distribution in the rectangular region [0, 2] × [0, 5]. This problem includes key concepts like expected value, double integration, and applying the magic product formula. Suitable for undergraduate students.

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