Final answer:
The student needs to calculate the four moments about the origin of a continuous random variable X with a given density function, by integrating the respective powers of X multiplied by the density function over the interval [0, 3].
Step-by-step explanation:
The student is asking to calculate the four moments about the origin for a continuous random variable X with a given density function f(x)=[4x(9−x²)]/81 within the interval [0, 3]. To find the moments about the origin, we need to calculate the expected values of X to the power of n, where n = 1, 2, 3, 4, using the formula for the n-th moment of a continuous random variable, \(\mu'_n = \int_{a}^{b} x^n f(x) dx\), where a and b are the bounds of the random variable, n is the moment number, and f(x) is the probability density function.
The first moment about the origin (the mean) is calculated as \(\mu'_1 = \int_{0}^{3} x f(x) dx\), the second moment is \(\mu'_2 = \int_{0}^{3} x^2 f(x) dx\), and similarly, the third and fourth moments would be found by integrating x^3 f(x) and x^4 f(x) over the interval from 0 to 3.