Question

If $X \sim N(\mu,\sigma^2)$ then:

• A. $P(X<\mu)=1$
• B. $P(X=\mu)=0$
• C. $P(X>\mu)=0$
• D. $P(X=\mu)=0.5$

Hint:

In continuous distributions, probability at a point is always zero.

Answer 1

The Correct Option is B

Concept: For a continuous random variable, probability at a single point is always zero: \[ P(X=a)=0 \] Normal distribution is a continuous distribution.

Step 1: Use property of continuous distribution.
Since $X$ is continuous, \[ P(X=\mu)=0 \]

Step 2: Check symmetry (additional insight).
Normal distribution is symmetric about mean: \[ P(X\mu)=\frac{1}{2} \] Final Conclusion: \[ P(X=\mu)=0 \]