Suppose that g is an easy probability density function to generate from, and h is a nonnegative function. Consider the following algorithm pseudo-code: Step 1. Generate Y ∼ g. Step 2. Generate E ∼ Exp(1) in the way that E = − log(U), U ∼ Unif(0, 1). Step 3. If E ≥ h(Y ), set X = Y . Otherwise go to Step 1. Step 4. Return X. This is a rejection algorithm and we want to find the density function of the generated samples. (d) Note that the density function f(x) in the samples is the conditional prob. f(x|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f(x) = c/x2 * e-(x^2)/2, x>1 (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)
Suppose that g is an easy
Step 1. Generate Y ∼ g.
Step 2. Generate E ∼ Exp(1) in the way that E = − log(U), U ∼ Unif(0, 1).
Step 3. If E ≥ h(Y ), set X = Y . Otherwise go to Step 1.
Step 4. Return X.
This is a rejection algorithm and we want to find the density function of the generated
samples.
(d) Note that the density function f(x) in the samples is the conditional prob. f(x|accepted).
Find f for X, subject to a constant.
(e) With the results, write the pseudo-code for the density
f(x) = c/x2 * e-(x^2)/2, x>1
(Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)
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