Given the problem: a joint density function for probability of three events is
given by the following equation:
f (x, y, z) ={(12x2yz  on [0, 1] × [0, 1] × [0, 1])(0 otherwise)

and the Previous question of: What is the probability that a chosen point (x, y, z) is such that 2x + y ≤ 1

Answer:
What is the expected value of each variable?

To find the expected value of each variable \( X \), \( Y \), and \( Z \) given the joint density function, we use the definition of the expected value for continuous random variables. The expected value of a random variable \( X \) is given by:

\[ E[X] = \iiint x f(x, y, z) \, dx \, dy \, dz \]

Similarly, the expected values for \( Y \) and \( Z \) are:

\[ E[Y] = \iiint y f(x, y, z) \, dx \, dy \, dz \]
\[ E[Z] = \iiint z f(x, y, z) \, dx \, dy \, dz \]

Given the joint density function \( f(x, y, z) \):

\[ f(x, y, z) = \begin{cases} 
12x^2yz & \text{if } 0 \leq x, y, z \leq 1 \\
0 & \text{otherwise}
\end{cases} \]

### Expected Value of \( X \)

\[ E[X] = \iiint_0^1 x \cdot 12x^2yz \, dx \, dy \, dz \]

Separate the integrals:

\[ E[X] = 12 \int_0^1 \int_0^1 \int_0^1 x^3 yz \, dx \, dy \, dz \]
\[ E[X] = 12 \left( \int_0^1 x^3 \, dx \right) \left( \int_0^1 y \, dy \right) \left( \int_0^1 z \, dz \right) \]

Compute each integral separately:

\[ \int_0^1 x^3 \, dx = \left[ \frac{x^4}{4} \right]_0^1 = \frac{1}{4} \]
\[ \int_0^1 y \, dy = \left[ \frac{y^2}{2} \right]_0^1 = \frac{1}{2} \]
\[ \int_0^1 z \, dz = \left[ \frac{z^2}{2} \right]_0^1 = 1 \]

Thus,

\[ E[X] = 12 \cdot \frac{1}{4} \cdot \frac{1}{2} \cdot 1 = \frac{12}{8} = \frac{3}{2} = 0.75 \]

### Expected Value of \( Y \)

\[ E[Y] = \iiint_0^1 y \cdot 12x^2yz \, dx \, dy \, dz \]

Separate the integrals:

\[ E[Y] = 12 \int_0^1 \int_0^1 \int_0^1 x^2 y^2 z \, dx \, dy \, dz \]
\[ E[Y] = 12 \left( \int_0^1 x^2 \, dx \right) \left( \int_0^1 y^2 \, dy \right) \left( \int_0^1 z \, dz \right) \]

Compute each integral separately:

\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \]
\[ \int_0^1 y^2 \, dy = \left[ \frac{y^3}{3} \right]_0^1 = \frac{1}{3} \]
\[ \int_0^1 z \, dz = \left[ \frac{z^2}{2} \right]_0^1 = 1 \]

Thus,

\[ E[Y] = 12 \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot 1 = \frac{12}{9} = \frac{4}{3} = 0.5 \]

### Expected Value of \( Z \)

\[ E[Z] = \iiint_0^1 z \cdot 12x^2yz \, dx \, dy \, dz \]

Separate the integrals:

\[ E[Z] = 12 \int_0^1 \int_0^1 \int_0^1 x^2 y z^2 \, dx \, dy \, dz \]
\[ E[Z] = 12 \left( \int_0^1 x^2 \, dx \right) \left( \int_0^1 y \, dy \right) \left( \int_0^1 z^2 \, dz \right) \]

Compute each integral separately:

\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \]
\[ \int_0^1 y \, dy = \left[ \frac{y^2}{2} \right]_0^1 = \frac{1}{2} \]
\[ \int_0^1 z^2 \, dz = \left[ \frac{z^3}{3} \right]_0^1 = \frac{1}{3} \]

Thus,

\[ E[Z] = 12 \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{1}{3} = \frac{12}{18} = \frac{2}{3} = 0.67 \]

### Summary

The expected values are:
- \( E[X] = 0.75 \)
- \( E[Y] = 0.5 \)
- \( E[Z] = 0.67 \)

Do you want details or have any questions?

Here are some more questions you might want to explore:
1. What is the variance of each variable \( X \), \( Y \), and \( Z \)?
2. What is the covariance between \( X \) and \( Y \)?
3. What is the probability that \( X + Y + Z \leq 1 \)?
4. How do the marginal density functions of \( X \), \( Y \), and \( Z \) look like?
5. Can you find the moment generating function for \( X \), \( Y \), and \( Z \)?

**Tip:** When calculating expected values, make sure to set up the correct limits of integration and factor the density function correctly for each variable.

Explore how to calculate the expected values of variables X, Y, and Z using the joint density function f(x, y, z). This problem involves integrating over the given joint density function to find the expected value for each variable. Suitable for advanced undergraduate students studying probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation