8.A random variable has a probability density function given by0 ≤ x ≤ 1,1 ≤ x ≤ 2,otherwise.f(x)={x - 1(This is the same PDF as in the previous question.) A computer simulation will be usedin which a sample of the random variable will need to be made. Determine and describe amethod to create a way to sample a random variable with the probability density functiongiven above.

The probability density function of the random variable is given by,The method to create a random sample from the given probability density function:1. Calculate the CDF:Find the cumulative distribution function (CDF) by integrating the probability density function (PDF). In this case, we have a piecewise PDF.The CDF of a random variable is defined as,For :For :2. Ensure :Evaluate the CDF at the upper limit of the integration, which is . Ensure that . If not, divide the entire CDF by to normalize it.For the given random variable, .Therefore further normalization is needed in this case.If , divide the entire CDF by to normalize it:This ensures that .The normalized CDF:Use the normalized CDF in inverse transform sampling.3. Inverse CDF:Find the inverse of the CDF, denoted as , where is a random sample from a uniform distribution in the range . This step is crucial for mapping uniform samples to the corresponding values of the random variable.The CDF of the random variable is defined as:The PDF can be inverted as (For simplicity denote the random variable as ):For ,For , Which implies,For quadratic equations the solution is given as,Therefore the solution to the above quadratic equation,If the value of the generated uniform sample is less than the value of is otherwise it is .4. Generate Random samples:Generate random samples from a uniform distribution . Apply the inverse CDF to obtain corresponding values of the random variable:These values constitute a sample from the desired distribution. The method to create a sample from the given pdf:Calculate the CDF.Ensure .Inverse the CDF.Generate random samples.