# 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.
$P ( X \gt x_0 ) = \;$
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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.
$P ( x_0 \lt X \lt x_1) = \;$
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