Question

Let \( X \) be a standard normal random variable. Then \( P(X^3 - 2X^2 - X + 2 > 0) \) equals

• A. \( 2\Phi(1) - 1 \)
• B. \( 1 - \Phi(2) \)
• C. \( 2\Phi(1) - \Phi(2) \)
• D. \( \Phi(2) - \Phi(1) \)

Hint:

When solving inequalities involving standard normal variables, it's often helpful to express the inequality in terms of the cumulative distribution function \( \Phi \).

Answer 1

The Correct Option is C

Step 1: Rewrite the expression. 
The given inequality \( X^3 - 2X^2 - X + 2 > 0 \) involves a cubic function of \( X \), which we solve numerically or use standard normal distribution tables to evaluate.

Step 2: Numerical approximation. 
By analyzing the roots of the cubic function and using the properties of the standard normal distribution, we find that: \[ P(X^3 - 2X^2 - X + 2 > 0) = 2\Phi(1) - \Phi(2). \]

Step 3: Conclusion. 
The correct answer is (C) \( 2\Phi(1) - \Phi(2) \).