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The probability density function for a normally distributed random variable \( X \) with mean \( \mu \) and standard deviation \( \sigma \) is given by:
\[ f_X(x,\mu,\sigma) = \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} \quad , \quad x \in \mathbb{R} \]
The probabilities that the random variable \( X \) is between, below or above certain values are given by the areas:
\[ P( x_0 \lt X \lt x_1 ) = \displaystyle \int_{x_0}^{x_1} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx \]
\[ P( X \lt x_0 ) = \displaystyle \int_{-\infty}^{x_0} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx \]
\[ P( X \gt x_0 ) = \displaystyle \int_{x_0}^{\infty} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx \]
There are no closed form solutions to the above integrals and they are therefore computed numerically.
We present three calculators that compute the three probabilities given above.
Enter the mean and standard deviation as real numbers; the standard deviation must be positive.
