Suppose it is relatively easy to simulate from the distributions Fi, i = 1, 2, . . . , n.

Question

Suppose it is relatively easy to simulate from the distributions F_i , i = 1, 2, . . . , n. If n is small, how can we simulate from

[latex]F(x) =\sum\limits^n_{i=1}P_i\, F_i(x), ~~~P_i\geqslant0 \quad \sum\limits_i P_i =1?[/latex]

Give a method for simulating from

[latex]F(x) =\begin{cases}\frac{1 − e^{−2x} + 2x}{3},\quad , 0 < x < 1\ \ \frac{3 − e^{−2x}}{3} \qquad\quad , 1 < x < ∞\end{cases}[/latex]

Accepted Answer

(a) Let U be a random number. If [latex]\sum_{j=1}^{i-1} P_{j}i. (In the preceding [latex]\sum_{j=1}^{i-1} P_{j} \equiv 0[/latex] when i=1.)

(b) Note that

[latex]F(x)=\frac{1}{3} F_{1}(x)+\frac{2}{3} F_{2}(x)[/latex]

where

[latex]\begin{aligned}& F_{1}(x)=1-e^{2 x}, \quad 0

Hence, using part (a), let U₁, U₂, U₃ be random numbers and set

[latex]X= \begin{cases}\frac{-\log U_{2}}{2}, & ext { if } U_{1}<\frac{1}{3} \ U_{3}, & ext { if } U_{1}>\frac{1}{3}\end{cases}[/latex]

The preceding uses the fact that −log U₂/2 is exponential with rate 2.

Question 11.EX.1: Suppose it is relatively easy to simulate from the distribut......