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me puedes hacer este ejercicio, con 5 decimales

The problem asks for the average yield of kilometers per liter on the road, given a joint probability density function \( f(x, y) \) where \( X \) represents kilometers in the city, and \( Y \) represents kilometers on the road.

The probability density function (PDF) is given by:

\[
f(x, y) = \frac{1}{4675} (2xy + x), \quad \text{for} \quad 6 \leq x \leq 11 \quad \text{and} \quad 8 \leq y \leq 13
\]

To find the average kilometers per liter on the road, you need to calculate the expected value of \( Y \), denoted as \( E[Y] \). The expected value for \( Y \) is calculated by the following formula:

\[
E[Y] = \int_{x=6}^{11} \int_{y=8}^{13} y \cdot f(x, y) \, dy \, dx
\]

First, we’ll set up and calculate the integral. Let me compute that for you.The expected value for \( Y \) (the average kilometers per liter on the road) is approximately \( 10.68939 \).

Thus, the average performance on the road is \( 10.68939 \), which matches the third option in the multiple-choice list.

Would you like more details, or do you have any other questions?

Here are 5 related questions to expand on this:

1. How would you compute the expected value for \( X \), the kilometers in the city?
2. What is the significance of the limits \( 6 \leq x \leq 11 \) and \( 8 \leq y \leq 13 \) in this problem?
3. How would the result change if the joint probability density function were different?
4. Can the expected value of \( X \) be interpreted similarly to \( Y \) in real-world terms?
5. How does the double integral calculation reflect the overall behavior of the vehicle's efficiency?

**Tip:** When calculating expected values from joint probability distributions, always ensure the proper integration limits for both variables are set based on the given support of the function.

Se sabe que la cantidad de kilómetros que rinde un auto es una variable aleatoria, donde X corresponde a los kilómetros rendidos en ciudad y Y los kilómetros rendidos en carretera por litro de combustible. La función de densidad de probabilidad es la siguiente:

f(x,y) = (1/4675) * (2xy + x), para 6 ≤ x ≤ 11 y 8 ≤ y ≤ 13.

Determine el rendimiento promedio en carretera.

Este ejercicio involucra la función de densidad de probabilidad de una variable aleatoria para determinar el rendimiento promedio en carretera. Utilizando un cálculo de valor esperado mediante una integral doble.

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