Step 1: Concept Identification: This question assesses foundational knowledge of the Cumulative Distribution Function (CDF) for a standard normal variable Z, characterized by a mean of 0 and a standard deviation of 1.
Step 2: Detailed Analysis: Evaluating each statement:
(A) \(F(Z) = \int_{-\infty}^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty<Z<\infty\): This equation defines the CDF of a standard normal distribution. It quantifies the area under the standard normal curve from \(-\infty\) up to a specific value Z. This statement is correct. (Correction: The exponent in the OCR was adjusted to the standard form \(-z^2/2\)).
(B) \(F(-Z) = 1 - F(Z)\): The standard normal distribution exhibits symmetry around its mean of 0. The CDF \(F(-Z)\) denotes the probability \(P(z \le -Z)\). Due to this symmetry, it is equivalent to the probability \(P(z \ge Z)\). We know that \(P(z \ge Z) = 1 - P(z<Z) = 1 - F(Z)\). This statement is correct.
(C) \(F(0) = 0\): \(F(0)\) represents the area under the standard normal curve to the left of Z=0. Given the curve's symmetry around 0, the area to the left of 0 constitutes precisely half of the total area, which is 1. Therefore, \(F(0) = 0.5\). This statement is incorrect.
(D) \(F(\infty) = 1\): The CDF evaluated at positive infinity, \(F(\infty)\), signifies the cumulative probability across the variable's entire range, i.e., \(P(Z \le \infty)\). The total area under any probability density function must sum to 1. This statement is correct.
Step 3: Conclusion: Statements (A), (B), and (D) accurately describe properties of the standard normal CDF. Consequently, the correct selection must include solely these three statements.