How are probabilities for the normal random variable represented?

Probabilities for a normal random variable are represented as areas under the curve of the normal distribution. The normal distribution is a continuous probability distribution that is symmetrical around its mean. The total area under the entire curve is equal to one, representing the total probability of all possible outcomes. Each specific probability associated with a range of values corresponds to the area under the curve for that interval. For instance, to find the probability that a normally distributed random variable falls within a certain range, one would calculate the area under the curve between the two values.

In contrast, representing probabilities as lengths of intervals does not accurately reflect the continuous nature of a normal distribution, as probabilities are not confined to a finite length but instead encompass areas. Ratios of mean and median do not provide a measure of probability, and while probabilities can be expressed as numerical values, this representation alone does not convey the continuous nature of the variable and its distribution characteristics. Thus, the correct interpretation aligns with the fundamental property of the normal distribution where probabilities are quantified as areas under the curve.