Consider a Weibull distribution with density

and assume that you can sample from a one$-$sided normal distribution with a parame$-$

ter $>0.$ Recall that the probability density function for a one$-$sided normal distribution

with the parameter $>0$ is given by

We aim to generate a sample from the Weibull distribution given above using the

Acceptance$-$rejection Method.

For each $>1,$ show that

$c()$:$=su{p}_{x}0f\frac{x}{g}(x)=\frac{\sqrt[\phantom{2}]{}}{2}{e}^{-\frac{1}{2}}(\frac{1}{2}-\frac{1}{2{}^2}{)}^{-\frac{1}{2}}.$

Show the value of $$ such that the one$-$sided normal distribution yields the most effi$-$

cient Acceptance$-$rejection scheme is $=\sqrt[\phantom{2}]{2}$

c$)$ Hence, calculate the acceptance rate for the most efficient Acceptance$-$rejection

scheme.

Give an advantage of using the one$-$sided normal distribution over the two$-$sided normal

distribution as the proposal distribution for Acceptance$-$rejection scheme to generate

a sample from the Weibull distribution.

Suggest an improvement if only a sample from two$-$sided normal distribution is avail$-$

able. Explain your answer.