Question

In the standard normal distribution, P(Z $\le$ 0) is equal to:
1. 1
2. 0.5
3. >0
4. <1

• A. 1
• B. 0.5
• C. $>$0
• D. $<$1

Hint:

For any symmetric continuous distribution centered at the origin, the median is \( 0 \), meaning exactly \( 50% \) of the distribution lies below \( 0 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question concerns the fundamental probability properties of the Standard Normal Distribution, commonly denoted as \( Z \sim N(0, 1) \).

Step 2: Detailed Explanation:
The standard normal distribution is a continuous probability distribution with a mean (\( \mu \)) of \( 0 \) and a standard deviation (\( \sigma \)) of \( 1 \).
The probability density function (PDF) of a standard normal distribution is defined mathematically as:
\[ \phi(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \] A primary characteristic of this distribution is its perfect symmetry about its mean, \( z = 0 \).
Because it is a continuous probability distribution, the total area under the probability density curve is exactly equal to \( 1 \):
\[ \int_{-\infty}^{\infty} \phi(z) \, dz = 1 \] Due to the symmetry about \( z = 0 \), the area under the curve is divided into two identical halves:
1. The area to the left of the mean, representing \( P(Z \le 0) \).
2. The area to the right of the mean, representing \( P(Z \ge 0) \).
Mathematically, this symmetry implies:
\[ P(Z \le 0) = P(Z \ge 0) = 0.5 \] Therefore, the probability of a standard normal random variable being less than or equal to \( 0 \) is precisely \( 0.5 \).

Step 3: Final Answer:
The value of \( P(Z \le 0) \) is \( 0.5 \).
Therefore, the correct choice is Option (B).