Problem $2($a$)[2$ points$]$

Consider the random variable $Y$ with density $g(y)=sin(y),$ with support in the interval $0,\frac{}{2}.$ Explain

how you can use inverse transformation method to generate values from this random variable. Describe the

method specifically with the functions given in this problem. You won't get any points for describing the

method in general terms. No R code needed here.

Problem $2($b$)[3$ points$]$

Write an R code to generate $10,000$ values from the random variable $Y$ using the inverse transformation method

that you described in part $($a$).$ Use seed $2024.$ Draw a histogram of the values you obtain, superimposed by a

smoothed density, using the following $R$ code $[Y$ is a vector of generated values in the following code$]$:

hist $,$ freq $=$ FALSE, breaks $=50$

curve$(sin(x),0,$ pi$/2,$ type $="$l$",$ add $=$ TRUE, col$=$"red"$)$

Show your $R$ code. Hint: On one of the boundaries of the distribution, the generated values might not look

quite right due to rounding error. Ignore issue due to rounding errors.

Problem $2($c$)[5$ points$]$

Write an $R$ code to implement the accept$-$reject algorithm that uses the random values of $Y$ that you generated

in part $($b$),$ to generate random values from the random variable $x$ with the density $f(x)=8\frac{x}{{}^2}$ with

support in $0,\frac{}{2}.$ To obtain an envelope obtain the optimal value of $$ such that $g\frac{y}{?}f(x)..$ Show how

you find $.$ Also show how many of the $10,000$ values that you generated are accepted. Note: There is no

need to use loops. Implement your code in vectorized form. Show your R code. Use the following code to

show your generated values superimposed by the density $f(x).$

hist $,$ freq $=$ FALSE, breaks $=50$

curve$(8*$x$/$pi$"2,0,$ pi$/2,$ type $="1",$ add $=$ TRUE, col$=$"red"$)$