Understanding the Fundamental Property: The Total Area under the Curve of a Normal Probability Distribution Always Adds up to 1

The area under the curve of a normal probability distribution is always equal to 1

The area under the curve of a normal probability distribution always adds up to 1

The area under the curve of a normal probability distribution always adds up to 1.

A normal probability distribution, also known as a bell curve or Gaussian distribution, is a continuous probability distribution defined by its mean (μ) and standard deviation (σ). The curve is symmetric and bell-shaped.

The total area under the curve represents the total probability of all possible outcomes in the distribution. Since probability is a measure that ranges from 0 to 1, the area under the curve must add up to 1.

This concept follows from the fundamental property of probability that the sum of probabilities of all possible outcomes in a sample space must equal 1. In the case of a normal distribution, the sample space is infinite, but the integral (area under the curve) from negative infinity to positive infinity always sums up to 1.

Mathematically, the area under the curve of a normal distribution is given by integrating the probability density function (PDF) over its entire range. The PDF of a standard normal distribution (with μ=0 and σ=1) is denoted as φ(x).

The integral of φ(x) from negative infinity (−∞) to positive infinity (∞) is:

∫[−∞, ∞] φ(x) dx = 1

This means that the total area under the curve of the standard normal distribution (μ=0 and σ=1) is equal to 1.

For any other normal distribution with a different mean (μ) and standard deviation (σ), the same property holds. The value of the standard deviation affects the spread or width of the curve, but the total area under the curve remains 1.

In summary, the area under the curve of a normal probability distribution always equals 1, providing a foundation for understanding and calculating probabilities within the distribution.

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