Notes on Inverse transform sampling

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

FY(a)=p(Y≤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.

FY(a)=p(F−1(U)≤a)

as F is monotonically increasing we could write,

FY(a)=p(U≤F(a))

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

FY(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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