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We are given a probability density function (PDF) for a random variable \( X \), and we are asked to find the value of \( E(Y^2) \), where \( Y = X^2 \).

### Step 1: Recall the formula for \( E(Y^2) \)
Since \( Y = X^2 \), we can express \( E(Y^2) \) as:

\[
E(Y^2) = E((X^2)^2) = E(X^4)
\]

Thus, we need to calculate \( E(X^4) \).

### Step 2: Use the formula for the expected value of a function of a random variable
The expected value of a function of a random variable, say \( g(X) \), is given by:

\[
E(g(X)) = \sum_x g(x) f(x)
\]

In this case, \( g(X) = X^4 \). We will compute \( E(X^4) \) using the probability distribution provided in the table. Let's list the probabilities:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & \frac{1}{10} \\
-1 & \frac{2}{10} \\
0 & \frac{1}{10} \\
1 & \frac{1}{10} \\
2 & \frac{2}{10} \\
3 & \frac{1}{10} \\
4 & \frac{2}{10} \\
\hline
\end{array}
\]

### Step 3: Calculate \( E(X^4) \)
We now compute the value of \( X^4 \) for each \( x \) and multiply it by the corresponding probability:

\[
E(X^4) = (-2)^4 \cdot \frac{1}{10} + (-1)^4 \cdot \frac{2}{10} + (0)^4 \cdot \frac{1}{10} + (1)^4 \cdot \frac{1}{10} + (2)^4 \cdot \frac{2}{10} + (3)^4 \cdot \frac{1}{10} + (4)^4 \cdot \frac{2}{10}
\]

Now calculate:

\[
E(X^4) = 16 \cdot \frac{1}{10} + 1 \cdot \frac{2}{10} + 0 \cdot \frac{1}{10} + 1 \cdot \frac{1}{10} + 16 \cdot \frac{2}{10} + 81 \cdot \frac{1}{10} + 256 \cdot \frac{2}{10}
\]

Simplifying:

\[
E(X^4) = \frac{16}{10} + \frac{2}{10} + \frac{0}{10} + \frac{1}{10} + \frac{32}{10} + \frac{81}{10} + \frac{512}{10}
\]

\[
E(X^4) = \frac{16 + 2 + 0 + 1 + 32 + 81 + 512}{10} = \frac{644}{10} = 64.4
\]

### Final Answer:
The value of \( E(Y^2) = E(X^4) = 64.4 \).

Since none of the options match this exact result, the correct choice is **b. none of these**.

---

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the expected value \( E(X) \) for the given probability distribution?
2. How would you compute the variance \( \text{Var}(X) \) of this random variable?
3. If \( Y = X^3 \), how would you calculate \( E(Y^2) \) for this new random variable?
4. How does the shape of the probability distribution affect the value of \( E(X^4) \)?
5. Can you compute \( E(X^2) \) for this distribution?

**Tip:** When dealing with functions of random variables, always check if the problem requires you to transform the variable (like \( X^4 \) in this case) before calculating expectations.

Let Y = X^2, then find the value of E(Y^2) based on the given probability density function.

Learn how to compute the expected value of Y^2, where Y = X^2, given a probability density function of X. This solution covers probability concepts, the expected value formula, and detailed step-by-step calculations of E(X^4). Suitable for advanced high school or undergraduate students.

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