Minimum of Two Exponential

Suppose we want to simulate pseudo random numbers from the following distribution, which I came across in r - Finding a way to simulate random numbers for this distribution - Cross Validated,

$$F(x) = 1-\exp\left(-ax - \frac{b}{p+1}x^{p+1}\right)\,\quad x\ge 0$$

where $a, b>0$ and $p\in (0,1)$.

Xi'an provided a very elegant solution.

Note that

$$(1-F(x))=\exp\left\{-ax-\frac{b}{p+1}x^{p+1}\right\}=\underbrace{\exp\left\{-ax\right\}}_{1-F_1(x)}\underbrace{\exp\left\{-\frac{b}{p+1}x^{p+1}\right\}}_{1-F_2(x)}$$

the distribution $F$ is the distribution of

$$X=\min\{X_1,X_2\}\qquad X_1\sim F_1\,,X_2\sim F_2$$

since

$$F(x) = P(X\le x) = 1- P(X > x) = 1-P(X_1 > x)P(X_2 > x) = 1-(1-F_1(X))(1-F_2(X))\,.$$

Thus, the R code to simulate is simple to be

x = pmin(rexp(n, a), rexp(n, b/(p+1))^(1/(p+1)))

References

• Finding a way to simulate random numbers for this distribution
• simulation fodder for future exams