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: Understanding the Concept:
This question tests 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:
Let's evaluate each statement:
(A) \(F(Z) = \int_{-\infty^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty<Z<\infty\):}
This is the mathematical definition of the CDF of a standard normal distribution. It represents the area under the standard normal curve from \(-\infty\) up to a specific value Z. This statement is correct. (Note: The OCR for the exponent was corrected to the standard form \(-z^2/2\)).
(B) \(F(-Z) = 1 - F(Z)\):
The standard normal distribution is symmetric about its mean, which is 0. The CDF \(F(-Z)\) represents the probability \(P(z \le -Z)\). Due to symmetry, this is equal 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) \(F(0) = 0\):
\(F(0)\) is the area under the standard normal curve to the left of Z=0. Since the curve is symmetric about 0, the area to the left of 0 is exactly half of the total area. The total area is 1, so \(F(0) = 0.5\). This statement is incorrect.
(D) \(F(\infty) = 1\):
The CDF evaluated at positive infinity, \(F(\infty)\), represents the cumulative probability over the entire range of the variable, i.e., \(P(Z \le \infty)\). The total area under any probability density function must be equal to 1. This statement is correct.
Step 3: Final Answer:
Statements (A), (B), and (D) are correct properties of the standard normal CDF. Therefore, the correct option includes only these three statements.