Question

A random variable \(X\) has a cumulative distribution function \(F_X(x)\) given by 

\( F_X(x) = \begin{cases} 0 & \text{if } x < 1 \\ \frac{x^{2} - 2x + 2}{2} & \text{if } 1 \le x < 2 \\ 1 & \text{if } x \ge 2 \end{cases} \)

Then \(E(X)\) is:

• A. $\frac{3}{2}$
• B. $\frac{4}{3}$
• C. $\frac{7}{3}$
• D. $\frac{5}{6}$

Hint:

For non-negative variables, $E(X) = \int_{0}^{\infty} [1 - F(x)] dx$.
$E(X) = \int_{0}^{1} (1-0) dx + \int_{1}^{2} (1 - \frac{x^2-2x+2}{2}) dx = 1 + \int_{1}^{2} \frac{2x - x^2}{2} dx$.
$1 + \frac{1}{2} [x^2 - x^3/3]_1^2 = 1 + \frac{1}{2} [(4 - 8/3) - (1 - 1/3)] = 1 + \frac{1}{2} [4/3 - 2/3] = 1 + 1/3 = 4/3$. This "complementary CDF" method handles discrete jumps automatically and is often less prone to error.

Answer 1

The Correct Option is B