Inverse Transfrom Method

• If X is cont. r.v with cdf F, then $F(x) \sim U[0,1]$
• Generate u from $U[0,1]$, then $Y = F−1(u)$ is a realization from F

Acceptance-Rejection Algorithm

• by sampling candidated from an easier dist
• then correcting the sampling probability by randomly rejecting some candidates
• If we have the reslut plot like that, we may accpet the red cases

x1 <- runif(300, 0, 1); y1 <- runif(300, 0, 2.6)
selected <- y1 < dbeta(x1, 3, 6)

plot(c(0,1), c(0,3), ty="n", main="A Sample Distribution", 
     ylab="Density f(x)", xlab="x")
curve (dbeta(x, 3, 6), add=TRUE)
lines(c(0,0,1,1), c(0,3,3,0))
points (x1, y1, col=1+1*selected, cex=0.1)

mean(selected)
accepted.points <- x1[selected]

# proportion of sample points less than 0.5
mean(accepted.points < 0.5) 

# The true distribution.
pbeta(0.5, 3, 6)

$\int^{0.5}_0 f(x)dx = 0.855$

Formally of Acceptance-Rejection Algorithm

• we know $f$, but we need to simulate from $f$.
• $g(x)$: easy to separet from distribution
  • $Y\sim g$
• envelope: $e(.)$
  • $e(.)$ is not a pdf
  • is a funcion that bounds my pdf
  • $e(x)=g(x)/\alpha \ge f(x)$
    • exceeds $f(x)$
    • $0\le \alpha \le 1$

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$Y \sim g ,\space\space U \sim unif(0,1)$