Important Properties of Normal Distribution

E

A normal distribution is symmetrical about the mean, which can be non-zero. It has a zero third moment (zero skewness).

The correct answer is E. I only.

Explanation:

I. It has only one mode: A normal distribution has a single mode, which is the value that occurs most frequently in the distribution. In a normal distribution, the mode, median, and mean are all equal. Therefore, statement I is true.

II. It has a positive third moment: The third moment of a distribution is a measure of its skewness. Skewness indicates the degree of asymmetry in the distribution. For a normal distribution, the third moment is zero, which means it is neither positively nor negatively skewed. Therefore, statement II is false.

III. It is symmetrical about zero: A normal distribution is symmetric, which means that the distribution is identical on both sides of its central point. In a normal distribution, the central point is the mean, and if the mean is zero, the distribution is symmetric about zero. Therefore, statement III is true.

To summarize, statements I and III are true, while statement II is false. Therefore, the correct answer is E. I only.