Question

IN R PROGRAMMING LANGUAGE. (Do not use functions in the code.)

Draw a sample with 1000 random observations from the distribution with probability density below for values 0 ≤ x ≤ 1:
Probability density -> f(x) = (4/3) x (2 − x 2 )
Show that the mean and variance of these 1000 random observation values are approximately equal to the mean and variance of the distribution whose probability density is given above.

Answer 1

A student is tasked with sampling 1000 random observations from a specific probability density function in R and verifying that the sample mean and variance match the theoretical values. The solution involves generating the sample without built-in functions and calculating the mean and variance for comparison.

Step-by-step explanation:

Sampling from a Probability Density Function in R

To address the student's question, we need to sample 1000 random observations from a given probability density function (pdf) without using built-in R functions. The probability density in question is
`f(x) = (4/3) * x * (2 - x^2)`, which is defined on the interval from 0 to 1. After generating the sample, we shall calculate the sample mean and variance and compare these to the theoretical mean and variance of the distribution. By the Law of Large Numbers and Central Limit Theorem, as the number of random observations increases, the mean and variance of the sample should closely approximate that of the distribution.

Sample Mean and Variance Calculation

Once the sample is drawn, the sample mean is calculated by summing all sampled values and dividing by the sample size (n), and the sample variance is computed by summing the squared differences from the sample mean, divided by the sample size minus one.

Theoretical calculations for the mean and variance of the probability density function involve integration over the defined range, which will not be performed manually here as it requires more advanced calculus.