Notes on Inverse transform sampling

Let \(F_Y(a)\) be the CDF of the r. v. \(Y\) and \(F\) be the CDF of the r. v. we want to sample.

\[F_Y(a) = p( Y \leq a)\]

Let us set \(Y\) to be equal to \(F^{-1} (U)\) where \(U\) is the uniform r. v. between 0 and 1. Now we can substitute the relation ship inthe equation as follows.

\[F_Y(a) = p( F^{-1}(U) \leq a)\]

as \(F\) is monotonically increasing we could write,

\[F_Y(a) = p( U \leq F(a))\]

As \(U\) is a r. v. following uniform distribution between 0 and 1, the above equation simplifies to,

\[F_Y(a) = F(a)\]

What this shows is that, the CDF of a random variable \(Y\), where \(Y\) is defined to be equal to \(F^{-1} (U)\) has the same CDF as \(F\).



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