Question

If \(X \sim N(\mu, \sigma^2)\). The maximum ordinate is at \(X = \mu\) and is given by:

• A. \(\sigma \sqrt{2\pi}\)
• B. \(\frac{1}{\sigma \sqrt{2\pi}}\)
• C. \(\sigma \sqrt{2\pi} e\)
• D. \(\frac{1}{\sigma \sqrt{2\pi e}}\)

Hint:

The peak value of any probability density function is its maximum ordinate.
For a standard normal distribution (\(\mu=0, \sigma=1\)), this peak value simplifies directly to \(\frac{1}{\sqrt{2\pi}} \approx 0.3989\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A normal distribution, denoted as \(X \sim N(\mu, \sigma^2)\), is a continuous probability distribution characterized by its bell-shaped curve.
The curve is symmetric about the mean \(\mu\), which is also the mode and median of the distribution.
The height of the curve at any point \(X\) is represented by its probability density function (PDF), also known as the ordinate.

Step 2: Key Formula or Approach:
The Probability Density Function (PDF) of a normally distributed random variable \(X\) is:
\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^2}, \quad -\infty < x < \infty \]

Step 3: Detailed Explanation:
The maximum value of the PDF (the peak of the bell curve) represents the maximum ordinate of the distribution.
Mathematically, the exponential term \(e^{-u}\) is maximized when the exponent \(-u\) is as close to zero as possible, since \(e^0 = 1\) and \(e^{-u} < 1\) for all \(u > 0\).
Therefore, the term \(e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^2}\) reaches its maximum value of 1 when:
\[ \frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^2 = 0 \implies x = \mu \]
Substituting \(x = \mu\) into the PDF expression, the exponential term becomes \(e^0 = 1\):
\[ f(\mu) = \frac{1}{\sigma \sqrt{2\pi}} e^{0} \]
\[ f(\mu) = \frac{1}{\sigma \sqrt{2\pi}} \]
This value, \(\frac{1}{\sigma \sqrt{2\pi}}\), is the maximum ordinate of the normal distribution curve and is located exactly at the mean \(X = \mu\).
This matches the second option.

Step 4: Final Answer:
Therefore, the correct option is (B).