What is a function that represents the distribution of a random variable as a symmetrical bell-shaped graph?

The normal distribution is indeed the correct answer, as it is characterized by its symmetrical, bell-shaped curve that represents how the values of a random variable are distributed. In a normal distribution, most of the observations cluster around the central peak, with probabilities tapering off symmetrically as you move away from the mean in either direction. This characteristic makes the normal distribution a foundational concept in statistics, as many natural phenomena tend to follow this pattern, notably heights, test scores, and other variables.

The importance of the normal distribution lies in its properties, such as the empirical rule, which states that approximately 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations. This makes it a critical tool for inferential statistics and hypothesis testing.

In contrast, other distributions like logarithmic, cubic, and geometric distributions each have distinct shapes and properties that do not conform to the symmetrical bell shape of the normal distribution. For example, a logarithmic distribution often skews to the right and is typically used for modeling phenomena such as income distribution, whereas cubic distributions are polynomial and not typically associated with randomness. Geometric distributions, on the other hand, relate