 Solution

For $0<u<1$ define the generalized inverse, or quantile function, byIt is finite because the cumulative distribution function tends to zero and one at the two infinities. Monotonicity and right continuity of $F$ give the key equivalenceFor the forward implication, choose points in the defining set approaching its infimum from above. Right continuity gives $F(Q(u))\ge u$, and then monotonicity gives $F(x)\ge u$ whenever $Q(u)\le x$. The reverse implication follows immediately because $x$ then belongs to the defining set. Consequently, for a continuous uniform distribution draw $U$,This proves inverse transform sampling for discontinuous as well as continuous distribution functions.

There is a minor endpoint defect in the printed codomain. The true quantile endpoint at probability one is $Q(1)=\inf \{x:F(x)=1\}$, which equals $+\infty$ for an unbounded-above normal distribution. Thus it need not be real at one. Use the extended-real value there, or assign an arbitrary finite value at this single probability level. Since $P(U=1)=0$, either choice gives the same sampled distribution; the latter supplies a real-valued version on all of $(0,1]$. The endpoint has no effect on the inverse-transform law.

For rejection sampling, choose a proposal probability density function $g$ that can be sampled and a finite constant $M$ with $f(y)\le Mg(y)$ wherever the target has positive mass. Generate independent pairs $(Y_j,U_j)$, with $Y_j$ of density $g$ and $U_j$ uniform. Accept $Y_j$ if $U_j\le f(Y_j)/(Mg(Y_j))$; otherwise discard the pair and repeat with fresh independent draws. At points where both densities vanish the acceptance rule may be defined arbitrarily. For any measurable set $A$,Thus one trial accepts with probability $1/M$, and the conditional density given acceptance is $f$. More explicitly, summing over the index of the first accepted proposal givesThe probability of never accepting is zero, since $M<\infty$, and the expected number of proposals is $M$. The algorithm terminates almost surely and samples the target density exactly.