Understanding Normal Probability Distribution Characteristics

The question asks to identify the true characteristics of a Normal probability distribution from the given statements. Let's analyze each statement:

Statement A: The two tails of normal curve extend to infinity in both directions and never touch horizontal axis.

This statement is true. A defining feature of the Normal probability distribution is that its theoretical curve, often called the bell curve, is asymptotic to the horizontal axis. This means the tails approach the axis infinitely closely but never actually touch it, extending indefinitely in both positive and negative directions.

Statement B: Normal curve is symmetrical around vertical line.

This statement is true. The Normal distribution curve is perfectly symmetrical. The vertical line of symmetry passes through the center of the distribution, which corresponds to the mean, median, and mode.

Statement C: The values of mean, median and mode are equal.

This statement is true. Due to the perfect symmetry of the Normal distribution, the peak of the curve occurs at the exact center. This central point represents the mean (average), the median (middle value), and the mode (most frequent value). Therefore, for a Normal distribution, $ \text{Mean} = \text{Median} = \text{Mode} $.

Statement D: Mean and variance of the distribution are equal.

This statement is generally false. While it is possible for the numerical value of the mean ($\mu$) to be equal to the numerical value of the variance ($ \sigma^2 $) for a specific Normal distribution (e.g., if $ \mu = 4 $ and $ \sigma^2 = 4 $), it is not a universal characteristic. The mean and variance are independent parameters that define the location and spread of the distribution, respectively. There is no inherent rule stating they must be equal.

Statement E: Mean and standard deviation are equal.

This statement is generally false. Similar to statement D, the mean ($ \mu $) and the standard deviation ($ \sigma $) can be numerically equal for certain specific Normal distributions (e.g., if $ \mu = 5 $ and $ \sigma = 5 $), but this is not a fundamental characteristic. The standard deviation represents the spread or dispersion of the data around the mean, and its value is not inherently tied to the value of the mean itself.

Conclusion

Based on the analysis, statements A, B, and C accurately describe the characteristics of a Normal probability distribution. Statements D and E are not general characteristics.

Therefore, the correct option includes A, B, and C only.