91Ó°ÊÓ

We first need to find the cumulative distribution function (CDF) of the given density function \(f(x)\). The CDF is the integral of the density function: \[F(x) = \int_0^x f(t) dt\] Now let's compute the CDF using the given density function: \[F(x) = \int_0^x 30(t^2 - 2t^3 + t^4) dt\] Now find the indefinite integral: \[F(x) = 30 \int (t^2 - 2t^3 + t^4) dt = 30\left(\frac{1}{3}t^3 - \frac{1}{2}t^4 + \frac{1}{5}t^5\right) + C\] Now, substitute the limits of integration: \[F(x) = 30\left(\frac{1}{3}x^3 - \frac{1}{2}x^4 + \frac{1}{5}x^5\right)\]

Next, find the inverse of the CDF, that is, the function \(F^{-1}(y)\) such that \(F(F^{-1}(y)) = y\). This step is required to use the inversion method for simulating the random variable. Since the inverse function cannot be found analytically, we will skip this step and proceed with Step 3 for a numerical solution.

To generate a random sample from the given distribution, we will use the inversion method. For this, we need to do the following: 1. Generate a random number \(\textit{u}\) from a uniform distribution in the range \([0,1]\). 2. Compute \(F^{-1}(u)\) numerically using an iterative or optimization method such as the Newton-Raphson method or the bisection method. This value is a random sample from the given distribution. Repeat step 1 and 2 for the desired number of random samples. In summary, to simulate a random variable with the given density function, we computed its CDF, then used the inversion method to generate random samples. Although the inverse of the CDF could not be obtained analytically, numerical methods can still be applied to compute random samples from the distribution.