Inverse Transform Sampling (Inverse CDF Sampling)

Q: What is inverse CDF sampling?

Inverse CDF sampling (also called inverse transform sampling) is a method for generating random numbers from a target distribution by plugging uniform random numbers between 0 and 1 into the inverse of that distribution's cumulative distribution function. For a normal distribution, this means using InvNorm (Φ⁻¹) on a uniform random number to produce a normally distributed random value.

Q: Why does inverse transform sampling work?

It works because of the probability integral transform: if X has cumulative distribution function F, then F(X) is uniformly distributed on (0,1). Reversing this, if U is uniform on (0,1), then F⁻¹(U) has the same distribution as X. Applying Φ⁻¹ (InvNorm) to a uniform random number therefore produces a normally distributed value.

Q: Is inverse transform sampling used in real software?

Yes. Many statistical libraries use inverse transform sampling or closely related methods to generate random variates, including R's qnorm-based generators and Python's scipy.stats distributions. It is a general-purpose technique that works for any distribution with a computable inverse CDF, not just the normal distribution.

Last reviewed: July 2026 by the InvNorm Calculator Editorial Team. Report an issue

Inverse transform sampling (also called inverse CDF sampling) is the standard method for generating random numbers from any distribution when you can compute the inverse of its cumulative distribution function. For the normal distribution, this inverse function is exactly the InvNorm function (Φ⁻¹) used throughout this site — which is why random-number generation is a natural extension of the same InvNorm calculation used to find Z scores and critical values.

The Idea Behind Inverse CDF Sampling

Every continuous random variable has a cumulative distribution function (CDF) that maps values to probabilities between 0 and 1. Inverse transform sampling reverses this: it starts from a probability and asks "what value produces this probability?" — exactly the question InvNorm answers for the normal distribution.

The method relies on the probability integral transform: if a random variable X has CDF F, then F(X) is uniformly distributed on the interval (0, 1). Reversing the transform, if U is a uniform random number on (0, 1), then F⁻¹(U) has exactly the same distribution as X.

Inverse Transform Sampling Algorithm

Generate a uniform random number u between 0 and 1 (most programming languages provide this directly, e.g. random() or Math.random()).
Apply the inverse CDF of the target distribution to u.
For the normal distribution, this step is InvNorm: z = Φ⁻¹(u).
The result z is a normally distributed random value with mean 0 and standard deviation 1. For a custom distribution, convert with x = μ + z × σ.

Worked Example

Suppose a random number generator produces u = 0.7291. Applying InvNorm:

z = Φ⁻¹(0.7291) ≈ 0.6103

Repeating this process with many different uniform random numbers produces a full sample of values that follow the standard normal distribution. Verify this value in the InvNorm calculator.

Inverse Transform Sampling in Code

Language	Code
Python (SciPy)	`from scipy.stats import norm; z = norm.ppf(random.random())`
R	`z <- qnorm(runif(1))`
Excel	`=NORM.S.INV(RAND())`

In practice, most statistical software provides a direct random-normal generator (such as numpy.random.normal() or R's rnorm()) that uses a similar or more optimized algorithm internally, but the inverse transform method underlies the concept and is straightforward to implement manually for any distribution with a known inverse CDF.

Where This Fits with InvNorm vs NormalCDF

Inverse transform sampling is a direct application of the same relationship covered in InvNorm vs NormalCDF: NormalCDF converts a value to a probability, while InvNorm converts a probability back to a value. Sampling simply supplies a random probability (drawn uniformly) as the input to InvNorm instead of a probability chosen by the user.

Inverse Transform Sampling FAQs

What is inverse CDF sampling?

Inverse CDF sampling (also called inverse transform sampling) is a method for generating random numbers from a target distribution by plugging uniform random numbers between 0 and 1 into the inverse of that distribution's cumulative distribution function. For a normal distribution, this means using InvNorm (Φ⁻¹) on a uniform random number to produce a normally distributed random value.

Why does inverse transform sampling work?

It works because of the probability integral transform: if X has cumulative distribution function F, then F(X) is uniformly distributed on (0,1). Reversing this, if U is uniform on (0,1), then F⁻¹(U) has the same distribution as X. Applying Φ⁻¹ (InvNorm) to a uniform random number therefore produces a normally distributed value.

Is inverse transform sampling used in real software?

Yes. Many statistical libraries use inverse transform sampling or closely related methods to generate random variates, including R's qnorm-based generators and Python's scipy.stats distributions. It is a general-purpose technique that works for any distribution with a computable inverse CDF, not just the normal distribution.

Inverse Transform Sampling (Inverse CDF Sampling)

The Idea Behind Inverse CDF Sampling

Inverse Transform Sampling Algorithm

Worked Example

Inverse Transform Sampling in Code

Where This Fits with InvNorm vs NormalCDF

Inverse Transform Sampling FAQs

Related Pages

InvNorm Calculator

InvNorm vs NormalCDF

Methodology