prove that area under normal distribution curve is 1
The area under a normal distribution curve can be proven to be 1 by using the properties of the probability density function of a normal distribution.
The probability density function of a normal distribution is given by the formula:
f(x) = (1/σ√(2π)) * e^(-(x-μ)^2 / (2σ^2))
Where:
- f(x) is the probability density function
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
To show that the total area under the normal distribution curve is equal to 1, we need to integrate the probability density function from negative infinity to positive infinity, which represents the entire range of the normal distribution.
∫[from -∞ to +∞] (1/σ√(2π)) * e^(-(x-μ)^2 / (2σ^2)) dx
This integral represents the area under the normal distribution curve. By evaluating this integral, it can be shown that the result is equal to 1.
Therefore, the total area under the normal distribution curve is indeed 1.