Question

Assertion (A): If \(X\sim N(\mu,\sigma^{2})\), then \[ P(X<\mu)=0.5. \] Reason (R): Normal distribution is symmetric about its mean.

• A. Both A and R are true and R is the correct explanation of A.
• B. Both A and R are true but R is not the correct explanation of A.
• C. A is true but R is false.
• D. A is false but R is true.

Hint:

For a normal distribution: \[ \boxed{ P(X\mu)=\frac12. } \] Also remember the important empirical rule: \[ \begin{aligned} P(\mu-\sigma P(\mu-2\sigma P(\mu-3\sigma<X<\mu+3\sigma) &\approx 99.7\%. \end{aligned} \] Whenever an Assertion-Reason question involves the symmetry of the normal distribution, first recall that the mean divides the total probability into two equal halves.

Answer 1

The Correct Option is A

Concept: A random variable following the normal distribution is denoted by \[ X\sim N(\mu,\sigma^{2}), \] where
• \(\mu\) is the mean of the distribution,
• \(\sigma^{2}\) is the variance,
• \(\sigma\) is the standard deviation. One of the most important properties of the normal distribution is that its probability density curve is perfectly symmetric about its mean \(\mu\). Due to this symmetry, \[ \boxed{ P(X\mu)=\frac12. } \] Thus, exactly half of the total area under the normal curve lies on each side of the mean.

Step 1: Examine the Assertion (A).
The assertion states that \[ P(X<\mu)=0.5. \] Since the normal distribution is symmetric about the mean, \[ \text{Area to the left of }\mu = \text{Area to the right of }\mu. \] The total probability under the curve is \[ 1. \] Hence, \[ P(X<\mu) = \frac12 = 0.5. \] Therefore, \[ \boxed{\text{Assertion (A) is True.}} \]

Step 2: Examine the Reason (R).
The reason states that \[ \text{``Normal distribution is symmetric about its mean.''} \] This is one of the fundamental properties of the normal distribution. The bell-shaped curve is divided into two identical halves by the vertical line \[ x=\mu. \] Hence, \[ \boxed{\text{Reason (R) is also True.}} \]

Step 3: Determine whether the Reason correctly explains the Assertion.
Since the normal distribution is symmetric about the mean, \[ \text{Left Area} = \text{Right Area}. \] Therefore, \[ P(X\mu) = \frac12. \] Thus, the statement in the Reason directly explains why \[ P(X<\mu)=0.5. \] Hence, \[ \boxed{\text{Reason (R) is the correct explanation of Assertion (A).}} \]

Step 4: Write the final answer.
Both the Assertion and the Reason are true, and the Reason correctly explains the Assertion. Therefore, \[ \boxed{\text{Option (A)}} \] is the correct answer.