Question

Let F(Z) be the cumulative density function of the standard normal variate Z, then which of the following are correct?
(A) \(F(Z) = \int_{-\infty}^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty < Z <\infty\)
(B) \(F(-Z) = 1 - F(Z)\)
(C) \(F(0) = 0\)
(D) \(F(\infty) = 1\)
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A), (B) and (C) only
• C. (A), (C) and (D) only
• D. (B) and (D) only

Hint:

For the standard normal distribution, remember these key CDF values: \(F(-\infty) = 0\), \(F(0) = 0.5\), and \(F(\infty) = 1\). The symmetry property \(F(-z) = 1 - F(z)\) is also crucial and frequently tested.

Answer 1

The Correct Option is A

Step 1: Concept Understanding:
This question assesses the fundamental properties of the Cumulative Distribution Function (CDF) for a standard normal variable Z, which has a mean of 0 and a standard deviation of 1.

Step 2: Detailed Explanation:
Each statement is evaluated:
(A) \(F(Z) = \int_{-\infty}^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty<Z<\infty\):
This is the definition of the CDF for a standard normal distribution. It signifies the area under the standard normal curve from \(-\infty\) to Z. This statement is correct. (Note: The OCR for the exponent was corrected to \(-z^2/2\)).

(B) \(F(-Z) = 1 - F(Z)\):
The standard normal distribution is symmetrical around its mean of 0. The CDF \(F(-Z)\) represents \(P(z \le -Z)\). Due to symmetry, this equals \(P(z \ge Z)\). Since \(P(z \ge Z) = 1 - P(z<Z) = 1 - F(Z)\), this statement is correct.

(C) \(F(0) = 0\):
\(F(0)\) is the area to the left of Z=0 under the standard normal curve. Due to symmetry, this area is half of the total area. The total area is 1, thus \(F(0) = 0.5\). This statement is incorrect.

(D) \(F(\infty) = 1\):
The CDF at positive infinity, \(F(\infty)\), represents the cumulative probability for the entire range, i.e., \(P(Z \le \infty)\). The total area under any probability density function is 1. This statement is correct.

Step 3: Final Answer:
Statements (A), (B), and (D) correctly describe properties of the standard normal CDF. The correct option includes only these three statements.