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I am trying to use the Inverse Transform Method to generate samples from this equation 1/12 * cube root of x where 0<=x<8. I have tried following some of the tutorials on the site with no success, please help me. I am trying to revise for an end of semester examination. How do I use the steps below to generate samples from the equation. Pardon my equation expressed in words I do not know how to use the formatting tools provided by the site to format the expression

a) Let F(x), x E R denote any cumulative distribution function

b) Let F−1(y)F−1(y), y∈[0,1]y∈[0,1] denote the inverse function and define X=F−1(U)X=F−1(U), where U∼U(0,1)U∼U(0,1)

c) Generate u∼U(0,1) u∼U(0,1).

d) Set u=1−e−0.05xu=1−e−0.05x (CDF of Exponential)

e) Solving for x, yields: x=−10.05log(u)

TechGeek
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  • You ignore d) and e) which are specific to the exponential context, and instead solve $F(x)=u$ where $F$ is your CDF in your actual problem, which is to say $x^{4/3}/16$. (You can ignore the fact that $F$ is not equal to $x^{4/3}/16$ everywhere because the solution you want will be in $[0,8]$ anyway.) – Ian Jun 13 '22 at 15:06
  • Would appreciate it if you posted a well defined step by step answer – TechGeek Jun 13 '22 at 15:11
  • Given what you have already written, the only "math" step left to do is to solve $x^{4/3}/16=u$ which is not difficult to do. The rest is programming, which is somewhat off-topic here. – Ian Jun 13 '22 at 15:14
  • okay i solve for that for the rest of values between 0 and 8? – TechGeek Jun 13 '22 at 15:16
  • You solve it for $u \in [0,1]$, which will end up hitting every $x \in [0,8]$. – Ian Jun 13 '22 at 15:24
  • Okay, does this site support dark mode?I just checked my profile tab and its not there – TechGeek Jun 13 '22 at 15:27

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