Question

When using inverse transform sampling, what specific property must the cumulative distribution function of the target random variable possess to ensure the generated samples follow the intended distribution?

Accepted Answer

To ensure that inverse transform sampling produces samples following the intended distribution, the cumulative distribution function (CDF) must be non-decreasing and right-continuous. The CDF, denoted as F(x), represents the probability that a random variable X takes a value less than or equal to x. Because the inverse transform sampling method works by taking a random value u from a uniform distribution between zero and one and finding the value x such that F(x) equals u, the function must be defined such that its inverse, the quantile function, is well-defined. If the CDF is strictly increasing, it has a unique inverse that maps every probability between zero and one to a single value in the domain of the distribution. If the CDF is not strictly increasing, such as when a distribution has flat regions where the probability density is zero, the inverse is defined as the generalized inverse, or infimum, which identifies the smallest value of x such that F(x) is at least u. Right-continuity ensures that the generalized inverse correctly maps the uniform random variable to the intended outcome space, preserving the underlying probability measure of the target distribution.

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When using inverse transform sampling, what specific property must the cumulative distribution function of the target random variable possess to ensure the generated samples follow the intended distribution?