To address the given problem, let's proceed with the tasks step-by-step, using a Cartesian coordinate system, which is suitable here because the limits are straightforwardly defined on $[0, 1] \times [0, 1] \times [0, 1]$.

1. Verifying the Probability Density Function

To verify that $f(x, y, z) = 12x^2yz$ is a probability density function, we need to ensure that the total integral over the domain equals 1:

[ \iiint_{[0,1] \times [0,1] \times [0,1]} 12x^2yz , dx , dy , dz ]

Setting up the integral:

$\iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} 12x^2yz \, dx \, dy \, dz$

2. Probability that $2x + y \leq 1$

To find the probability that a chosen point $(x, y, z)$ satisfies $2x + y \leq 1$, we need to integrate $f(x, y, z)$ over the region where $2x + y \leq 1$:

$\iiint_{D} 12x^2yz \, dx \, dy \, dz$

where $D$ is the region defined by $0 \leq x \leq 1$, $0 \leq y \leq 1 - 2x$, and $0 \leq z \leq 1$.

Setting up the integral:

$\iiint_{0}^{1} \iiint_{0}^{1 - 2x} \iiint_{0}^{1} 12x^2yz \, dz \, dy \, dx$

3. Expected Value of Each Variable

The expected value of a random variable $X$ is given by:

[ E[X] = \iiint_{[0,1] \times [0,1] \times [0,1]} x \cdot f(x, y, z) , dx , dy , dz ]

Similarly, the expected values for $Y$ and $Z$ are:

[ E[Y] = \iiint_{[0,1] \times [0,1] \times [0,1]} y \cdot f(x, y, z) , dx , dy , dz ]

[ E[Z] = \iiint_{[0,1] \times [0,1] \times [0,1]} z \cdot f(x, y, z) , dx , dy , dz ]

Setting up the integrals:

• $E[X]$:

$E[X] = \iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} x \cdot 12x^2yz \, dx \, dy \, dz$

• $E[Y]$:

$E[Y] = \iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} y \cdot 12x^2yz \, dx \, dy \, dz$

• $E[Z]$:

$E[Z] = \iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} z \cdot 12x^2yz \, dx \, dy \, dz$

Summary of Integrals

1. Verifying the Probability Density Function:

$\iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} 12x^2yz \, dx \, dy \, dz$

2. Probability that $2x + y \leq 1$:

$\iiint_{0}^{1} \iiint_{0}^{1 - 2x} \iiint_{0}^{1} 12x^2yz \, dz \, dy \, dx$

3. Expected Values:

• $E[X]$:

$E[X] = \iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} 12x^3yz \, dx \, dy \, dz$

• $E[Y]$:

$E[Y] = \iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} 12x^2y^2z \, dx \, dy \, dz$

• $E[Z]$:

$E[Z] = \iiint_{0}^{1} \iiint_{0}^{1} \iiint_{0}^{1} 12x^2yz^2 \, dx \, dy \, dz$

Do you want any details on the integration process or have any questions?

Here are five additional questions you might consider asking:

1. How do you perform these integrals step-by-step?
2. What are the interpretations of the expected values in probability?
3. How would the setup change if the limits of integration were different?
4. Can you explain how the region $2x + y \leq 1$ is determined geometrically?
5. How can these integrals be evaluated using numerical methods?

Tip: When dealing with joint density functions, visualizing the region of integration can greatly help in setting up the correct limits.