Ask AI: Why is the area under a bell-shaped curve equal to one?

The area under a bell-shaped curve, also known as the Gaussian distribution or normal distribution, is equal to one because it represents the probability distribution of a continuous random variable. In other words, the area under the curve signifies the total probability of all possible outcomes for that variable. In the context of a normal distribution, the curve is determined by the mean and standard deviation of the underlying data. The curve represents the probability density function (PDF) of that distribution. The key characteristics of a normal distribution are: 1. It is symmetrical around its mean (center). 2. The area under the curve represents the probabilities of the possible outcomes. 3. The mean, median, and mode are equal and occur at the peak of the curve. 4. Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Since the total probability of all possible outcomes in a probability distribution must add up to 1 (or 100%), the area under the bell-shaped curve must equal one. This concept illustrates the continuous nature of probability distribution – the integration of the PDF across the entire range of values will always result in one.