What is Inverse Normal Distribution?

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

In other words, the way to work backward for finding the x-value using the known probability is referred to as an inverse normal distribution.

Example: For stock market returns and prices, a characteristic is that it models the extremely large variations from (crashes) that can occur even when almost all (normal) variations are small.

How to use an Inverse Normal Distribution?

The probability density function for the inverse normal distribution is given by:

$f\left(x,\mu ,\lambda \right)=\sqrt{\frac{\lambda }{2\pi x^3}}{e}^{\left(-\frac{\lambda {\left(x-\mu \right)}^2}{2{\mu }^{2}x}\right)}$

The function is defined if x>0, where $\mu >0$ is the mean and $\lambda >0$ is the shape parameter.

The inverse normal distribution always works on the left tail. For finding the value for inverse normal distribution, the inverse normal distribution table is generally used. To use the inverse normal distribution table, the area under the curve, the mean, and the variance should be known.

For example:

If P(X ≤ x)=0.2 and $X\sim N\left(88,{19}^2\right)$, find the value of x.

By using the inverse normal distribution table,

$f^{-1}\left(0.2,88,19\right)=72.0092$

By rounding the value, x=72.

The inverse normal distribution will not work on the right tail. In that case, if the shaded area is given for the right tail then consider the unshaded area inside the curve and then find the value of the inverse normal distribution.

For example:

If $X\sim N\left(88,{19}^2\right)$ and the probability that X is greater than Q is 0.25, find the value of Q.

The area of the unshaded region is 1–0.25=0.75.

By using the inverse normal distribution table,

$f^{-1}\left(0.75,88,19\right)=100.815$

By rounding the value, x=101.

Mean

In statistics, the mean is the average of numbers. It is a measure of the central tendency of a probability distribution. There are so many different ways to calculate the mean.

The mean of the inverse normal distribution:

$E[X]=\mu$

Variance

Variance is a measure of how the data set is spread out. It is calculated as the average squared deviation of each number from the mean of a data set.

The variance of the inverse normal distribution is $Var[X]={\sigma }^2=\frac{{\mu }^3}{\lambda }$.

Standard Deviation

Standard deviation is a square root of variance.

The standard deviation of the inverse normal distribution is $S.D=\sigma =\sqrt{\frac{{\mu }^3}{\lambda }}$.

Z-score

A z-score can be measured using the standard deviation and mean. A z-score is always a numerical value. A z-score may be positive or negative. If it is positive, the score will be above the mean and if it is negative, the score will be below the mean. A z-score is also called a standard score.

For example, if the z-score is zero then the score of the data point is identical to the mean score.

Formula for Standard Score

$z=\frac{x-\mu }{\sigma }$, where x denotes the score, $\mu$ denotes the mean and $\sigma$ denotes the standard deviation.

Continuous Distribution

The random variable X that can take any value between a given range is said to be following continuous probability distribution. For example, the inverse normal distribution is a continuous probability distribution with a family of two parameters.

Density function

The probability density function is also known as the density function. The density function is used to denote the probability distribution function for continuous random variables. The density function is nonnegative and it’s integral over the entire space is equal to one.

Exponential Distribution

The exponential distribution is also called the negative exponential distribution. The exponential distribution is a probability distribution that describes the time between events in a Poisson point process. In exponential distribution, the failure rate will not be constant for modelling the technical devices.

For example, the amount of time until a natural hazard occurs has an exponential distribution.

Formula for Exponential Distribution

$f(x)=\frac{1}{\beta }{e}^{\frac{-(x-\mu )}{\beta }}$, $x\ge \mu ;\beta >0$

Here $\mu$ represents the location parameter and $\beta$ represents the scale parameter.

The scale parameter is referred to as $\lambda$ is equal to $\frac{1}{\beta }$.

For $\mu =0;\beta =1$, the exponential distribution is known as the standard exponential distribution.

Inverse Exponential Distribution

The inverse exponential distribution is used for the modelling of datasets with the inverted bathtub failure rates. The inverse exponential distribution is applied for describing real-life events in medicine, engineering, and biology.

Confidence Interval

The confidence interval provides the range of the values for a parameter, which is unknown. Usually, the confidence intervals are plotted as graphs or reported in table format along with the estimated points of the same parameters. For calculating the confidence intervals, the inverse normal distribution is used.

Inverse Gaussian Distribution

The inverse Gaussian distribution is also known as Wald distribution. The inverse Gaussian distribution has some properties similar to the Gaussian distribution. The Gaussian distribution describes the level of the Brownian motion at a fixed time while the inverse Gaussian describes the distribution of time that a Brownian motion with positive drift takes to reach the level of fixed position.

The inverse Gaussian distribution is also called the normal-inverse Gaussian distribution. The name inverse Gaussian was used by Tweedie due to the inverse relationship between time and distance. It provides unity for both mean and scale. It is the standard form for all distributions. The tail of inverse Gaussian distribution decreases more slowly in comparison with the normal distribution. Therefore, inverse Gaussian distribution is more suitable for large values model and it forms a subclass of the generalized hyperbolic distributions. 

Generalized Inverse Gaussian (GIG) Distribution

The GIG distribution was introduced by Good. The GIG distribution is applied in many fields such as finance, statistical linguistics, geostatistics, and it is also widely used for modeling and analyzing lifetime data and soon.

Inverse Weibull (IW) Distribution

The Inverse Weibull (IW) distribution is also known as the reciprocal Weibull distribution. IW distribution is used to describe the degradation phenomena of mechanical components. This distribution has many important applications such as life testing, useful life, infant mortality rate.

Numerical Analysis of Inverse Normal Distribution

Example:

The weights of 12-year old students’ backpacks are normally distributed with a mean 3 kg and a standard deviation 0.5 kg. Above what weight would 15% of the backpacks lie?

Mean and Standard deviation:

$\begin{array}{l}\mu =3\\ \sigma =0.5\end{array}$

Consider that the weight above W that would fall into the category of 15%.

By using the normal distribution table, the area between mean and variance will be 0.35.

Using the value 0.35, the probability of the area being 0.35 is 1.036.

That is z=1.036.

The formula of the standard score is $z=\frac{x-\mu }{\sigma }$, where x denotes the score, $\mu$ denotes the mean and $\sigma$ denotes the standard deviation.

Substitute x=W, $\mu =3$, $\sigma =0.5$ and z=1.036 in $z=\frac{x-\mu }{\sigma }$,

$\begin{array}{l}1.036=\frac{W-3}{0.5}\\ W-3=\left(1.036\right)\left(0.5\right)\\ W-3=0.518\\ W=3+0.518\end{array}$

W=3.518kg

Hence, 15% of the backpacks will lie above the weight of 3.518 kg.

Advantages of Inverse Normal Distribution

The inverse normal distribution is used for calculating the value of z for the given area below a certain value, above a certain value, between two values, or outside two values.

Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for      

• Bachelor of Science in Statistics
• Master of Science in Statistics