# Inverse Normal Probability Calculator

A calculator that calculates the random variable given the normal probability is presented; this is the to finding the probability given the random variable.

Recall that 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 varibale \( X \) is between, below or above certain values are given by

\[ P( X \lt x_0 ) = \displaystyle \int_{-\infty}^{x_0} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx \] \[ P( X \gt x_0 ) = \displaystyle \int_{x_0}^{\infty} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx \] \[ P( x_0 \lt X \lt x_1 ) = \displaystyle \int_{x_0}^{x_1} \dfrac{1}{\sigma \sqrt{2 \pi}} e^{ - \dfrac{(x-\mu^2)}{2 \sigma^2}} dx \] This calcultor solves the inverse problem: given the probability find the random variable \( X \) for all three possibilities above.

We present three calculators that compute the random variable given the probability \( P_0 \) such that \( 0 \le P_0 \le 1\).

Mean , Standard Deviation =

Decimal Places =1) Find \( x_0 \) such that \( P( X \lt x_0 ) = P_0 \). Enter \( P_0 \) in the text area below.

\( P ( X \lt x_0 ) = \; \) ,

2) Find \( x_0 \) such that \( P( X \gt x_0 ) = P_0 \). Enter \( P_0 \) in the text area below.

\( P ( X \gt x_0 ) = \; \) ,

2) Find \( x_0 \) and \( x_1 \)such that \( P ( x_0 \lt X \lt x_1) = P_0 \). Enter \( P_0 \) in the text area below. Note that the intreval \( [ x_0 , x_1] \) is center around the mean.

\( P ( x_0 \lt X \lt x_1) = \; \) ,