Question

Determine a method to generate random observations for the following pdf: $$ f(x)=\left\\{\begin{array}{ll} 4 x^{3} & 0

Short answer

The R function which generates \'n\' random observations from the given pdf is:\[\begin{verbatim} generate_random <- function(n) { y <- runif(n) x <- y^(1/4) return(x)} \end{verbatim} \]

Step-by-step solution

Identify the PDF and its domain

The given probability density function (pdf) is \( f(x) = 4x^3 \) in the domain \( 0 < x < 1 \), and \( f(x) = 0 \) elsewhere. This tells us that the random variable can only take values between 0 and 1.

Computing the Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) can be computed by integrating the pdf over its domain. In our case, the CDF \( F(x) = \int_0^x f(u) du = \int_0^x 4u^3 du = u^4 \) evaluated from 0 to x, which results in \( F(x) = x^4 \) for \( 0 < x < 1 \), and 0 or 1 elsewhere.

Implementing Inverse Transform Sampling

The inverse transform sampling method requires the inverse of the CDF, \( F^{-1}(x) \). This can be computed by solving \( F(x) = y \) for \( x \), where \( y \) is a uniformly distributed random variable in (0,1). From \( F(x) = x^4 \) we find \( F^{-1}(y) = \sqrt[4]{y} \) for \( 0 < y < 1 \). This is the transformation we apply to the uniformly distributed random variates to obtain the desired distribution.

Write the R function

The R function for generating random observations from the given pdf can be written as follows: \[\begin{verbatim} generate_random <- function(n) { y <- runif(n) x <- y^(1/4) return(x)} \end{verbatim} \]The function takes one argument, \( n \), the number of random observations required. The function generates \( n \) uniformly distributed random numbers in (0,1) using the runif function, then transforms these numbers using the inverse CDF \( \sqrt[4]{\cdot} \), and returns the transformed numbers.