An actuary is using the following model for the proportion of settled claims after time $t$: The proportion is $X$ where
$$P(X \leq x) = x^te^{t(1-x)}$$
Time is measured in small units. This is not a stochastic processes model; the actuary is just modeling the marginal distribution of the proportion of settled claims at a fixed time. Generate samples from the distribution of $X$ for various values of $t$.
Continuing form the previous question, a mathematician discovered that the random variable $Z = t(\log(X))^2$ is approximately exponentially distributed with mean $2$ for large values of $t$. She did that by calculating the limit of $P(Z > z)$ as $t \to \infty$. The actuary was sceptical about the approximation overall and in particular, she claimed that the approximation was performing especially badly for large values of $Z$ (corresponding to small values of $X$). Investigate all arguments using a sample generated in the previous question and suggest an improvement by choosing a different approximating distribution.
I think writing a Matlab code is the way to go about with this problem. But I am not very well versed with Matlab to understand how to extract a sample and what the range of $t$ should be for the first part of the problem.