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Inverse Normal Distribution a.k.a Inverse Guassian Function

Typical Question: Adult male population's height (in cm) can be modelled by a continuous random variable \(X\) that follows a normal distribution with mean \(\mu = 175\) and standard deviation \(\sigma = 7\); \(X \sim N \begin{pmatrix} 175 , 7^2 \end{pmatrix}\). What height is \(90\%\) of the adult male population shorter than? In other words: how tall is the top \(10\%\) of the adult male population?

This question is illustrated in the bell curve we see here. In essence, we're given an area under the bell curve and we need to find the value of the continuous random variable that area corresponds to.

Given a probability \(p\), which corresponds to an area under a bell curve, the inverse normal distribution function will allow us to calculate the value of \(x\) that \(X\) must be less than or equal to, \(X\leq x \), so that \(P\begin{pmatrix}X \leq x\end{pmatrix} = p\).

So for the typical question, shown above, the inverse normal function will allow us to calculate the value \(x = k\) such that: \[P\begin{pmatrix}X \leq k \end{pmatrix} = 0.9\] We'll refer to the inverse normal function using either of the following notations: \[F^{-1}(p) \quad \text{or} \quad invNorm(p)\] Notice that the input value for the inverse normal function is a probability (an area under the bell curve) and the output value is a value of \(x\).

The key differences between the normal distribution and the inverse normal distribution are highlighted here.

Normal Distribution

The normal distribution function \(F(x)\) allows us to calculate the probability that a continuous random variable \(X\), which follows a normal distribution, be less than or equal to a given value \(x\): \[F(x) = P\begin{pmatrix}X \leq x \end{pmatrix}\] In terms of the bell curve, the normal distribution tells us the area enclosed for a given value of \(x\):

Inverse Normal Distribution

The inverse normal distribution function \(F^{-1}(x)\) on the other hand allows us to calculate the value of \(x\), that \(X\) must be less than or equal to, for a given probability \(p\): \[F^{-1}(p) = x, \quad \text{such that:}\quad P\begin{pmatrix} X \leq x \end{pmatrix} = p\] In terms of the bell curve, the inverse normal distribution tells us the value of \(x\) for a given area (a given probability):

Tutorial: The Inverse Normal Distribution

In this tutorial we consider a continuous random variable that follows a normal distribution with mean \(\mu = 88\) and standard deviation \(\sigma = 19\).

Worked Example 1

Adult male's height, in France, follows a normal distribution with mean \(\mu = 175cm\) and standard deviation \(\sigma = 7cm\) .

  1. Find the height that \(97\%\) of the male population lies under.
  2. Benjamin claims to be within the tallest \(10\%\) of the population. What is the shortest he can be, for his claim to be true?

Solution

  1. We're told that \(97\%\) of the male population is shorter than the height we need to find. We'll call the unknown height \(k\).
    Since the area shaded, beneath the curve, is a Left Tail we can use the Inverse Normal Distribution Function and plug in the area (probability) of \(0.97\), with the parameters \(\mu = 175\) and \(\sigma = 7\), and it will return the value of \(k\): \[F^{-1}(0.97,175,7) = 188.166\] Rounding to \(3\) significant figures: \[F^{-1}(0.97,175,7) = 188\] This tells us that \(97\%\) of the adult male population are shorter than \(188\)cm.
    We can find this result with our calculator, using the built-in Inverse Normal Distribution function, invNorm, using the input value \(0.97\) and the parameters \(\mu = 175\) and \(\sigma = 7\): \[invNorm(0.97,175,7) = 188.166\]
    Rounding to \(3\) significant figures: \[invNorm(0.97,175,7) = 188\]
  2. In this case we're trying to find the
    This is illustrated on the bell curve we see here. We need to find the value \(k\) such that the area enclosed by the curve and the \(x\)-axis to the right hand side of \(k\) is equal to \(0.1\).
    IMPORTANT RULE: when using the inverse normal function, we can only input areas (probabilities) that start from the far left-hand side of the bell-curve; such areas are known as Left Tails.

    Put Simply: the Inverse Normal Distribution Function only Works with Left Tails.

    The rule we just read, tells us that because the area of \(0.1\) we're dealing with in this problem starts on the far right-hand side, is a Right Tail we need to consider the area to the left of \(k\) (the complement probability). Here's how it works:
    We use the fact that the area enclosed on the left hand side of \(k\) must be equal to: \[1 - 0.1 = 0.9\] We can see this on the second bell-curve shown here. The area of \(0.9\) is now a Left Tail and we can use the inverse normal function.
    Using the inverse normal function, with \(0.9\) as an input value, \(\mu = 175\) and \(\sigma = 7\) as parameters, will give us the value of \(k\).
    Using our calculator, we find: \[F^{-1}(0.9,175,7) = 183.971\] Rounding to the nearest unit: \[F^{-1}(0.9,175,7) = 184cm\] We could also write: \[invNorm(0.9,175,7)=184cm\]
    Finally, we can state that for Benjamin's claim, that he is within the tallest \(10\%\) of the population, to be true he must measure at least \(184\)cm.

Worked Example 2

A continuous random variable \(X\) follows a normal distribution, with mean \(\mu = 88\) and standard deviation \(\sigma = 19\), so that \(X\sim N \begin{pmatrix}88,19^2\end{pmatrix}\).

Find the value of:

  1. \(k\) such that \(P\begin{pmatrix} X \leq k \end{pmatrix}=0.6\)
  2. \(m\) such that \(P\begin{pmatrix} X \geq m \end{pmatrix}=0.2\)
  3. \(n\) such that \(P\begin{pmatrix} \mu \leq X \leq n \end{pmatrix}=0.15\)

Solution


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