Question 1172355
Let's find the expected value of X².

**Understanding Expected Value**

The expected value of a function g(X) of a continuous random variable X with probability density function (PDF) f(x) is given by:

E[g(X)] = ∫ g(x) * f(x) dx

**Applying the Formula**

In this case:

* g(X) = X²
* f(x) = 1 - x/2 for 0 ≤ x ≤ 2
* f(x) = 0 elsewhere

So, we need to calculate:

E[X²] = ∫(from 0 to 2) x² * (1 - x/2) dx

**Calculating the Integral**

1.  **Expand the expression:**
    * E[X²] = ∫(from 0 to 2) (x² - x³/2) dx

2.  **Integrate:**
    * E[X²] = [x³/3 - x⁴/8](from 0 to 2)

3.  **Evaluate the integral at the limits:**
    * E[X²] = [(2)³/3 - (2)⁴/8] - [(0)³/3 - (0)⁴/8]
    * E[X²] = [8/3 - 16/8] - [0]
    * E[X²] = 8/3 - 2
    * E[X²] = 8/3 - 6/3
    * E[X²] = 2/3

**Result**

Therefore, the expected value of X² is 2/3, which is approximately 0.6667.